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Research Article General-Appell Polynomials within the...
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Hindawi Publishing Corporation International Journal of Analysis Volume 2013, Article ID 328032, 11 pages http://dx.doi.org/10.1155/2013/328032 Research Article General-Appell Polynomials within the Context of Monomiality Principle Subuhi Khan and Nusrat Raza Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Correspondence should be addressed to Subuhi Khan; [email protected] Received 22 September 2012; Accepted 6 December 2012 Academic Editor: Jacques Liandrat Copyright © 2013 S. Khan and N. Raza. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A general class of the 2-variable polynomials is considered, and its properties are derived. Further, these polynomials are used to introduce the 2-variable general-Appell polynomials (2VgAP). e generating function for the 2VgAP is derived, and a correspondence between these polynomials and the Appell polynomials is established. e differential equation, recurrence relations, and other properties for the 2VgAP are obtained within the context of the monomiality principle. is paper is the first attempt in the direction of introducing a new family of special polynomials, which includes many other new special polynomial families as its particular cases. 1. Introduction and Preliminaries e Appell polynomials are very oſten found in different applications in pure and applied mathematics. e Appell polynomials [1] may be defined by either of the following equivalent conditions: { ()}( ∈ N 0 ) is an Appell set ( being of degree exactly ) if either, (i) (/) () = −1 () ( ∈ N 0 ) or (ii) there exists an exponential generating function of the form () exp () = ∞ ∑ =0 () ! , (1) where () has (at least the formal) expansion: () = ∞ ∑ =0 ! ( 0 / = 0) . (2) Roman [2] characterized Appell sequences in several ways. Properties of Appell sequences are naturally handled within the framework of modern classical umbral calculus by Roman [2]. We recall the following result [2, eorem 2.5.3], which can be viewed as an alternate definition of Appell sequences. e sequence () is Appell for (), if and only if 1 () exp () = ∞ ∑ =0 () ! , (3) where () = ∞ ∑ =0 ! ( 0 / = 0) . (4) In view of (1) and (3), we have () = 1 () . (5) e Appell class contains important sequences such as the Bernoulli and Euler polynomials and their generalized forms. Some known Appell polynomials are listed in Table 1. We recall that, according to the monomiality principle [15, 16], a polynomial set { ()} ∈N is “quasimonomial”, provided there exist two operators and playing, respectively, the role of multiplicative and derivative operators, for the family of polynomials. ese operators satisfy the following identities, for all ∈ N: { ()} = +1 () , (6) { ()} = −1 () . (7)
Transcript of Research Article General-Appell Polynomials within the...
Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2013 Article ID 328032 11 pageshttpdxdoiorg1011552013328032
Research ArticleGeneral-Appell Polynomials within theContext of Monomiality Principle
Subuhi Khan and Nusrat Raza
Department of Mathematics Aligarh Muslim University Aligarh 202002 India
Correspondence should be addressed to Subuhi Khan subuhi2006gmailcom
Received 22 September 2012 Accepted 6 December 2012
Academic Editor Jacques Liandrat
Copyright copy 2013 S Khan and N RazaThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A general class of the 2-variable polynomials is considered and its properties are derived Further these polynomials areused to introduce the 2-variable general-Appell polynomials (2VgAP) The generating function for the 2VgAP is derived anda correspondence between these polynomials and the Appell polynomials is established The differential equation recurrencerelations and other properties for the 2VgAP are obtained within the context of the monomiality principle This paper is the firstattempt in the direction of introducing a new family of special polynomials which includes many other new special polynomialfamilies as its particular cases
1 Introduction and Preliminaries
The Appell polynomials are very often found in differentapplications in pure and applied mathematics The Appellpolynomials [1] may be defined by either of the followingequivalent conditions 119860
119899(119909)(119899 isin N
0) is an Appell set (119860
119899
being of degree exactly 119899) if either
(i) (119889119889119909)119860119899(119909) = 119899119860
119899minus1(119909) (119899 isin N
0) or
(ii) there exists an exponential generating function of theform
119860 (119905) exp (119909119905) =
infin
sum119899=0
119860119899
(119909)119905119899
119899 (1)
where 119860(119905) has (at least the formal) expansion
119860 (119905) =
infin
sum119899=0
119860119899
119905119899
119899(1198600
= 0) (2)
Roman [2] characterized Appell sequences in severalways Properties of Appell sequences are naturally handledwithin the framework of modern classical umbral calculus byRoman [2] We recall the following result [2 Theorem 253]which can be viewed as an alternate definition of Appellsequences
The sequence 119860119899(119909) is Appell for 119892(119905) if and only if
1
119892 (119905)exp (119909119905) =
infin
sum119899=0
119860119899
(119909)119905119899
119899 (3)
where
119892 (119905) =
infin
sum119899=0
119892119899
119905119899
119899(1198920
= 0) (4)
In view of (1) and (3) we have
119860 (119905) =1
119892 (119905) (5)
TheAppell class contains important sequences such as theBernoulli and Euler polynomials and their generalized formsSome known Appell polynomials are listed in Table 1
We recall that according to themonomiality principle [1516] a polynomial set 119901
119899(119909)119899isinN is ldquoquasimonomialrdquo provided
there exist two operators and playing respectivelythe role of multiplicative and derivative operators for thefamily of polynomials These operators satisfy the followingidentities for all 119899 isin N
119901119899
(119909) = 119901119899+1
(119909) (6)
119901119899
(119909) = 119899119901119899minus1
(119909) (7)
2 International Journal of Analysis
Table1Listof
someA
ppellp
olyn
omials
SNo
119892(119905
)119860
(119905)
Generatingfunctio
nsPo
lyno
mials
(I)
119892(119905
)=
(119890119905
minus1)
119905119860
(119905)
=119905
(119890119905minus
1)
119905
(119890119905minus
1)119890119909119905
=infin sum119899=0
119861119899
(119909)
119905119899 119899
TheB
erno
ullipo
lyno
mials[3]
(II)
119892(119905
)=
(119890119905
+1)
2119860
(119905)
=2
(119890119905+
1)
2
(119890119905+
1)119890119909119905
=infin sum119899=0
119864119899
(119909)
119905119899 119899
TheE
uler
polyno
mials[3]
(III)
119892(119905
)=
(119890119905
minus1)120572
119905120572119860
(119905)
=119905120572
(119890119905minus
1)120572
119905120572
(119890119905minus
1)120572
119890119909119905
=infin sum119899=0
119861(120572)
119899(119909
)119905119899 119899
Theg
eneralized
Bernou
llipo
lyno
mials[4]
(IV)
119892(119905
)=
(119890119905
+1)120572
2120572
119860(119905
)=
2120572
(119890119905+
1)120572
2120572
(119890119905+
1)120572
119890119909119905
=infin sum119899=0
119864(120572)
119899(119909
)119905119899 119899
Theg
eneralized
Eulerp
olyn
omials[4]
(V)
119892(119905
)=
(1198901205721119905
minus1)
(1198901205722119905
minus1)
sdotsdotsdot(
119890120572119898119905
minus1)
12057211205722
sdotsdotsdot120572119898
119905119898
12057211205722
sdotsdotsdot120572119898
119905119898
(1198901205721119905minus
1)(1198901205722119905minus
1)sdotsdot
sdot (119890120572119898119905minus
1)119890119909119905
=infin sum119899=0
119861(119898)
119899(119909
|12057211205722
120572119898
)119905119899 119899
Theg
eneralized
Bernou
llipo
lyno
mialsof
order119898
[5]
119860(119905
)=
12057211205722
sdotsdotsdot120572119898
119905119898
(1198901205721119905minus
1)(1198901205722119905minus
1)sdotsdot
sdot (119890120572119898119905minus
1)
(VI)
119892(119905
)=
(1198901205721119905
+1)
(1198901205722119905
+1)
sdotsdotsdot(
119890120572119898119905
+1)
2119898
2119898
(1198901205721119905+
1)(1198901205722119905+
1)sdotsdot
sdot (119890120572119898119905+
1)119890119909119905
=infin sum119899=0
119864(119898)
119899(119909
|12057211205722
120572119898
)119905119899 119899
Theg
eneralized
Eulerp
olyn
omialsof
order119898
[5]
119860(119905
)=
2119898
(1198901205721119905+
1)(1198901205722119905+
1)sdotsdot
sdot (119890120572119898119905+
1)
(VII)
119892(119905
)=
119890119905minus
sum119898minus1
ℎ=0
(119905ℎ
ℎ)
119905119898
119860(119905
)=
119905119898
119890119905minus
sum119898minus1
ℎ=0
(119905ℎℎ
)
119905119898
119890119905minus
sum119898minus1
ℎ=0
(119905ℎℎ
)119890119909119905
=infin sum119899=0
119861[119898minus1]
119899(119909
)119905119899 119899
Then
ewgeneralized
Bernou
llipo
lyno
mials[6]
International Journal of Analysis 3
Table1Con
tinued
SNo
119892(119905
)119860
(119905)
Generatingfunctio
nsPo
lyno
mials
(VIII)
119892(119905
)=
(119905
120582119890119905
minus1
)minus120572
119860(119905
)=
(119905
120582119890119905
minus1
)120572
(119905
120582119890119905
minus1
)120572
119890119909119905
=infin sum119899=0
B(120572)
119899(119909
120582)
119905119899 119899
TheA
posto
l-Berno
ullipo
lyno
mialsof
order120572
[7]
(IX)
119892(119905
)=
120582119890119905
minus1
119905119860
(119905)
=119905
120582119890119905
minus1
119905
120582119890119905
minus1
119890119909119905
=infin sum119899=0
B119899
(119909120582
)119905119899 119899
TheA
posto
l-Berno
ullipo
lyno
mials[78]
(X)
119892(119905
)=
(2
120582119890119905
+1
)minus120572
119860(119905
)=
(2
120582119890119905
+1
)120572
(2
120582119890119905
+1
)120572
119890119909119905
=infin sum119899=0
E(120572)
119899(119909
120582)
119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheA
posto
l-Euler
polyno
mialsof
order120572
[79]
(XI)
119892(119905
)=
120582119890119905
+1
2119860
(119905)
=2
120582119890119905
+1
2
120582119890119905
+1
119890119909119905
=infin sum119899=0
E119899
(119909120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheA
posto
l-Euler
polyno
mials[7ndash9
]
(XII)
119892(119905
)=
119890minus(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
119860
(119905)
=119890(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
120585 119903+1
=0
119890(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
119890119909119905
=infin sum119899=0
119860119899
(119909)
119905119899 119899
Theg
eneralized
Gou
ld-H
opperp
olyn
omials[10]
(for
119903=
1the
Hermite
polyno
mials
119867119899(119909
)[11]andfor
119903=
2classical2-orthogon
alpo
lyno
mials)
(XIII)
119892(119905
)=
(1minus
119905 )119898+1
119860(119905
)=
1
(1minus
119905)119898+1
1
(1minus
119905)119898+1119890119909119905
=infin sum119899=0
119866(119898)
119899(119909
)119905119899
TheM
iller-Lee
polyno
mials[1112](for119898
=0the
trun
catedexpo
nentialp
olyn
omials
119890 119899(119909
)[11]andfor
119898=
120573minus
1the
mod
ified
Laguerre
polyno
mials
119891(120573)
119899(119909
)
[13])
(XIV
)119892
(119905)
=(119890119905
+1)
2119905
119860(119905
)=
2119905
(119890119905+
1)
2119905
(119890119905+
1)119890119909119905
=infin sum119899=0
119866119899
(119909)
119905119899 119899
TheG
enocchip
olyn
omials[14
]
4 International Journal of Analysis
The operators and also satisfy the commutationrelation
[ ] = minus = 1 (8)
and thus display the Weyl group structure If the consideredpolynomial set 119901
119899(119909)119899isinN is quasimonomial its properties
can easily be derived from those of the and operatorsIn fact
(i) Combining the recurrences (6) and (7) we have
119901119899
(119909) = 119899119901119899
(119909) (9)
which can be interpreted as the differential equationsatisfied by 119901
119899(119909) if and have a differential
realization(ii) Assuming here and in the sequel 119901
0(119909) = 1 then
119901119899(119909) can be explicitly constructed as
119901119899
(119909) = 119899
1199010
(119909) = 119899
1 (10)
which yields the series definition for 119901119899(119909)
(iii) Identity (10) implies that the exponential generatingfunction of 119901
119899(119909) can be given in the form
exp (119905) 1 =
infin
sum119899=0
119901119899
(119909)119905119899
119899(|119905| lt infin) (11)
We note that the Appell polynomials 119860119899(119909) are quasi-
monomial with respect to the following multiplicative andderivative operators
119860
= 119909 +1198601015840
(119863119909)
119860 (119863119909)
(12a)
or equivalently
119860
= 119909 minus1198921015840
(119863119909)
119892 (119863x) (12b)
119860
= 119863119909 (13)
respectivelyThe special polynomials of two variables are useful from
the point of view of applications in physics Also thesepolynomials allow the derivation of a number of useful iden-tities in a fairly straightforward way and help in introducingnew families of special polynomials For example Bretti etal [17] introduced general classes of two variables Appellpolynomials by using properties of an iterated isomorphismrelated to the Laguerre-type exponentials
We consider the 2-variable general polynomial (2VgP)family 119901
119899(119909 119910) defined by the generating function
119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899(1199010
(119909 119910) = 1) (14)
where 120601(119910 119905) has (at least the formal) series expansion
120601 (119910 119905) =
infin
sum119899=0
120601119899
(119910)119905119899
119899(1206010
(119910) = 0) (15)
We recall that the 2-variable Hermite Kampe de Ferietpolynomials (2VHKdFP) 119867
119899(119909 119910) [18] the Gould-Hopper
polynomials (GHP) 119867(119898)
119899(119909 119910) [19] and the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) [20] are defined by the
generating functions
119890119909119905+119910119905
2
=
infin
sum119899=0
119867119899
(119909 119910)119905119899
119899 (16)
119890119909119905+119910119905
119898
=
infin
sum119899=0
119867(119898)
119899(119909 119910)
119905119899
119899 (17)
119860 (119905) 119890119909119905+119910119905
2
=
infin
sum119899=0
119867119860119899
(119909 119910)119905119899
119899 (18)
respectively Thus in view of generating functions (14) (16)(17) and (18) we note that the 2VHKdFP 119867
119899(119909 119910) the GHP
119867(119898)
119899(119909 119910) and the HAP
119867119860119899(119909 119910) belong to 2VgP family
In this paper operational methods are used to introducecertain new families of special polynomials related to theAppell polynomials In Section 2 some results for the 2-variable general polynomials (2VgP) 119901
119899(119909 119910) are derived
Further the 2-variable general-Appell polynomials (2VgAP)119901119860119899(119909 119910) are introduced and framed within the context
of monomiality principle In Section 3 the Gould-Hopper-Appell polynomials (GHAP)
119867119860(119898)
119899(119909 119910) are considered
and their properties are established Some members belong-ing to theGould-Hopper-Appell polynomial family are given
2 2-Variable General-Appell Polynomials
In order to introduce the 2-variable general-Appell polyno-mials (2VgAP) we need to establish certain results for the2VgP 119901
119899(119909 119910) Therefore first we prove the following results
for the 2VgP 119901119899(119909 119910)
Lemma 1 The 2VgP 119901119899(119909 119910) defined by generating function
(14) where 120601(119910 119905) is given by (15) are quasimonomial underthe action of the following multiplicative and derivative opera-tors
119901
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
(1206011015840
(119909 119905) =120597
120597119905120601 (119909 119905)) (19)
119901
= 119863119909 (20)
respectively
Proof Differentiating (14) partially with respect to 119905 we have
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (21)
International Journal of Analysis 5
Table2Listof
someG
ould-H
opper-A
ppellp
olyn
omials119867
119860(119898)
119899(119909
119910)
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(I)
(119890119905
minus1)
119905
119905
(119890119905minus
1)
119909+
119898119910
119863119898minus1
119909+
((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905
(119890119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Be
rnou
llipo
lyno
mials
(II)
(119890119905
+1)
2
2
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909minus
119890119863119909
(119890119863119909
+1)119863119909
2
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898)
119899(119909
119910)
119905119899 119899
TheG
ould
Hop
per-Eu
ler
polyno
mials
(III)
(119890119905
minus1)120572
119905120572
119905120572
(119890119905minus
1)120572
119909+
119898119910
119863119898minus1
119909+
120572((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905120572
(119890119905minus
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mials
(IV)
(119890119905
+1)120572
2120572
2120572
(119890119905+
1)120572
119909+
119898119910
119863119898minus1
119909minus
120572119890119863119909
(119890119863119909
+1)119863119909
2120572
(119890119905+
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omials
(V)
119904 prod ℎ=1
(119890120572ℎ119905
minus1)
120572ℎ119905119904
119909
+119898
119910119863119898minus1
119909+
119898119863minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909
(119890120572119903119863119909
minus1)119863119909
1205721
sdotsdotsdot120572119904119905119904
(1198901205721119905minus
1)sdotsdot
sdot (119890120572119904119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mialsof
order119904
119904 prod ℎ=1
120572ℎ
(119890120572ℎ119905minus
1)119905119904
(VI)
119904 prod ℎ=1
(119890120572ℎ119905
+1)
2119904
119909
+119898
119910119863119898minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909119863119909
2119904
(1198901205721119905+
1)sdotsdot
sdot (119890120572119904119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omialsof
order119904
119904 prod ℎ=1
2119904
(119890120572ℎ119905+
1)
(VII)
119890119905minus
sum119904minus1
ℎ=0
(119905ℎ
ℎ)
119905119904
119909+
119898119910
119863119898minus1
119909+
119898119863minus1
119909minus
1minus
119863119909
119898minus1
(119898
minus1)
(119890119863119909
minussum119898minus1
ℎ=0
119905ℎℎ
)119863119909
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898[119904minus1])
119899(119909
119910)
119905119899 119899
TheG
ould-H
oppern
ewgeneralized
Bernou
llipo
lyno
mials
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)
6 International Journal of Analysis
Table2Con
tinued
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(VIII)
(119905
120582119890119905
minus1
)minus120572
(119905
120582119890119905
minus1
)120572
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(120572minus
119863119909)
minus120572
119863119909
(120582119890119863119909
minus1)
119863119909
(119905
120582119890119905
minus1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898120572)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mialsof
order120572
(IX)
120582119890119905
minus1
119905
119905
120582119890119905
minus1
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(1minus
119863119909)
minus1
119863119909
(120582119890119863119909
minus1)
119863119909
119905
120582119890119905
minus1
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mials
(X)
(2
120582119890119905
+1
)minus120572
(2
120582119890119905
+1
)120572
119909+
119898119910
119863119898minus1
119909minus
120582120572
119890119863119909
(120582119890119863119909
+1)119863119909
(2
120582119890119905
+1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898120572)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheG
ould-H
opper-A
posto
l-Euler
polyno
mialsof
order120572
(XI)
120582119890119905
+1
2
2
120582119890119905
+1
119909+
119898119910
119863119898minus1
119909minus
120582119890119863119909
(120582119890119863119909
+1)119863119909
2
120582119890119905
+1
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheG
ould-H
opper-A
posto
l-Euler
polyno
mials
(XII)
119890minus(sum119903+1
ℎ=0120585ℎ119905ℎ)
119890(sum119903+1
ℎ=0120585ℎ119905ℎ)
120585 119903+1
=0
119909+
119898119910
119863119898minus1
119909+119903+1
sumℎ=1
ℎ120585 ℎ
119863ℎminus1
119909119863119909
119890(sum119903+1
ℎ=0120585ℎ119905ℎ+119909119905+119910119905119898)
=infin sum119899=0
119867119860(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Gou
ld-H
opperp
olyn
omials(fo
r119903=
1
theG
ould-H
opper-Hermite
polyno
mials119867
119867(119898)
119899(119909
119910)andfor
119903=
2the
Gou
ld-H
opperc
lassical
2-orthogon
alpo
lyno
mials)
(XIII)
(1minus
119905 )119904+1
1
(1minus
119905 )119904+1
119909+
119898119910
119863119898minus1
119909minus
119898+
1
1minus
119863119909
119863119909
1
(1minus
119905 )119904+1119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898119904)
119899(119909
119910)
119905119899
TheG
ould-H
opper-Miller-Lee
polyno
mials(fo
r119904=
0the
Gou
ld-H
opper-trun
catedexpo
nential
polyno
mials119867
119890119898 119899(119909
119910)andfor
119904=
120573minus
1Gou
ld-H
opper-mod
ified
Laguerre
polyno
mials119867
119891(119898120573)
119899(119909
119910))
(XIV
)(119890119905
+1)
2119905
2119905
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909+
119890119863119909
(1minus
119863119909)
+1
(119890119863119909
+1)
119863119909
2119905
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Genocchi
polyno
mials
International Journal of Analysis 7
If 120601(119910 119905) is an invertible series and 1206011015840
(119910 119905)120601(119910 119905) hasTaylorrsquos series expansion in powers of 119905 then in view of theidentity
119863119909
119890119909119905
120601 (119910 119905) = 119905 (119890119909119905
120601 (119910 119905)) (22)
we can write
1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119890119909119905
120601 (119910 119905) =1206011015840
(119910 119905)
120601 (119910 119905)(119890119909119905
120601 (119910 119905)) (23)
Now using (23) in the lhs of (21) we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (24)
Making use of generating function (14) in the lhs of theabove equation we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899
(25)
which on equating the coefficients of like powers of 119905 in bothsides gives
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119901119899
(119909 119910) = 119901119899+1
(119909 119910) (26)
Thus in view of monomiality principle equation (6) theabove equation yields assertion (19) of Lemma 1 Again usingidentity (22) in (14) we have
119863119909
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119899minus1
(119909 119910)119905119899
(119899 minus 1) (27)
Equating the coefficients of like powers of 119905 in both sidesof (27) we find
119863119909
119901119899
(119909 119910) = 119899119901119899minus1
(119909 119910) (119899 ge 1) (28)
which in view of monomiality principle equation (7) yieldsassertion (20) of Lemma 1
Remark 2 The operators given by (19) and (20) satisfycommutation relation (8) Also using expressions (19) and(20) in (9) we get the following differential equation satisfiedby 2VgP 119901
119899(119909 119910)
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus 119899) 119901119899
(119909 119910) = 0 (29)
Remark 3 Since 1199010(119909 119910) = 1 therefore in view of monomi-
ality principle equation (10) we have
119901119899
(119909 119910) = (119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
119899
1 (1199010
(119909 119910) = 1) (30)
Also in view of (11) (14) and (19) we have
exp (119901119905) 1 = 119890
119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 (31)
Now we proceed to introduce the 2-variable general-Appell polynomials (2VgAP) In order to derive the gener-ating functions for the 2VgAP we take the 2VgP 119901
119899(119909 119910) as
the base in the Appell polynomials generating function (1)Thus replacing 119909 by the multiplicative operator
119901of the
2VgP 119901119899(119909 119910) in the lhs of (1) and denoting the resultant
2VgAP by119901119860119899(119909 119910) we have
119860 (119905) 119890(119901119905)
=
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (32)
Now using (31) in the exponential term in the lhs of(32) we get the generating function for
119901119860119899(119909 119910) as
119860 (119905) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (33)
In view of (5) generating function (33) can be expressedequivalently as
1
119892 (119905)119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (34)
Now we frame the 2VgAP119901119860119899(119909 119910) within the context
of monomiality principle formalism We prove the followingresults
Theorem 4 The 2VgAP119901119860119899(119909 119910) are quasimonomial with
respect to the following multiplicative and derivative operators
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
(35a)
or equivalently
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
minus1198921015840
(119863119909)
119892 (119863119909)
(35b)
119901119860
= 119863119909 (36)
respectively
Proof Differentiating (33) partially with respect to 119905 we find
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)+
1198601015840
(119905)
119860 (119905)) 119860 (119905) 119890
119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(37)
Since 119860(119905) and 120601(119910 119905) are invertible series of 119905 therefore1198601015840
(119905)119860(119905) and 1206011015840
(119910 119905)120601(119910 119905) possess power series expan-sions of 119905 Thus in view of identity (22) we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119860 (119905) 119890119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(38)
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
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Volume 2014
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International Journal of
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Operations ResearchAdvances in
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Analysis
Table1Listof
someA
ppellp
olyn
omials
SNo
119892(119905
)119860
(119905)
Generatingfunctio
nsPo
lyno
mials
(I)
119892(119905
)=
(119890119905
minus1)
119905119860
(119905)
=119905
(119890119905minus
1)
119905
(119890119905minus
1)119890119909119905
=infin sum119899=0
119861119899
(119909)
119905119899 119899
TheB
erno
ullipo
lyno
mials[3]
(II)
119892(119905
)=
(119890119905
+1)
2119860
(119905)
=2
(119890119905+
1)
2
(119890119905+
1)119890119909119905
=infin sum119899=0
119864119899
(119909)
119905119899 119899
TheE
uler
polyno
mials[3]
(III)
119892(119905
)=
(119890119905
minus1)120572
119905120572119860
(119905)
=119905120572
(119890119905minus
1)120572
119905120572
(119890119905minus
1)120572
119890119909119905
=infin sum119899=0
119861(120572)
119899(119909
)119905119899 119899
Theg
eneralized
Bernou
llipo
lyno
mials[4]
(IV)
119892(119905
)=
(119890119905
+1)120572
2120572
119860(119905
)=
2120572
(119890119905+
1)120572
2120572
(119890119905+
1)120572
119890119909119905
=infin sum119899=0
119864(120572)
119899(119909
)119905119899 119899
Theg
eneralized
Eulerp
olyn
omials[4]
(V)
119892(119905
)=
(1198901205721119905
minus1)
(1198901205722119905
minus1)
sdotsdotsdot(
119890120572119898119905
minus1)
12057211205722
sdotsdotsdot120572119898
119905119898
12057211205722
sdotsdotsdot120572119898
119905119898
(1198901205721119905minus
1)(1198901205722119905minus
1)sdotsdot
sdot (119890120572119898119905minus
1)119890119909119905
=infin sum119899=0
119861(119898)
119899(119909
|12057211205722
120572119898
)119905119899 119899
Theg
eneralized
Bernou
llipo
lyno
mialsof
order119898
[5]
119860(119905
)=
12057211205722
sdotsdotsdot120572119898
119905119898
(1198901205721119905minus
1)(1198901205722119905minus
1)sdotsdot
sdot (119890120572119898119905minus
1)
(VI)
119892(119905
)=
(1198901205721119905
+1)
(1198901205722119905
+1)
sdotsdotsdot(
119890120572119898119905
+1)
2119898
2119898
(1198901205721119905+
1)(1198901205722119905+
1)sdotsdot
sdot (119890120572119898119905+
1)119890119909119905
=infin sum119899=0
119864(119898)
119899(119909
|12057211205722
120572119898
)119905119899 119899
Theg
eneralized
Eulerp
olyn
omialsof
order119898
[5]
119860(119905
)=
2119898
(1198901205721119905+
1)(1198901205722119905+
1)sdotsdot
sdot (119890120572119898119905+
1)
(VII)
119892(119905
)=
119890119905minus
sum119898minus1
ℎ=0
(119905ℎ
ℎ)
119905119898
119860(119905
)=
119905119898
119890119905minus
sum119898minus1
ℎ=0
(119905ℎℎ
)
119905119898
119890119905minus
sum119898minus1
ℎ=0
(119905ℎℎ
)119890119909119905
=infin sum119899=0
119861[119898minus1]
119899(119909
)119905119899 119899
Then
ewgeneralized
Bernou
llipo
lyno
mials[6]
International Journal of Analysis 3
Table1Con
tinued
SNo
119892(119905
)119860
(119905)
Generatingfunctio
nsPo
lyno
mials
(VIII)
119892(119905
)=
(119905
120582119890119905
minus1
)minus120572
119860(119905
)=
(119905
120582119890119905
minus1
)120572
(119905
120582119890119905
minus1
)120572
119890119909119905
=infin sum119899=0
B(120572)
119899(119909
120582)
119905119899 119899
TheA
posto
l-Berno
ullipo
lyno
mialsof
order120572
[7]
(IX)
119892(119905
)=
120582119890119905
minus1
119905119860
(119905)
=119905
120582119890119905
minus1
119905
120582119890119905
minus1
119890119909119905
=infin sum119899=0
B119899
(119909120582
)119905119899 119899
TheA
posto
l-Berno
ullipo
lyno
mials[78]
(X)
119892(119905
)=
(2
120582119890119905
+1
)minus120572
119860(119905
)=
(2
120582119890119905
+1
)120572
(2
120582119890119905
+1
)120572
119890119909119905
=infin sum119899=0
E(120572)
119899(119909
120582)
119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheA
posto
l-Euler
polyno
mialsof
order120572
[79]
(XI)
119892(119905
)=
120582119890119905
+1
2119860
(119905)
=2
120582119890119905
+1
2
120582119890119905
+1
119890119909119905
=infin sum119899=0
E119899
(119909120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheA
posto
l-Euler
polyno
mials[7ndash9
]
(XII)
119892(119905
)=
119890minus(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
119860
(119905)
=119890(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
120585 119903+1
=0
119890(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
119890119909119905
=infin sum119899=0
119860119899
(119909)
119905119899 119899
Theg
eneralized
Gou
ld-H
opperp
olyn
omials[10]
(for
119903=
1the
Hermite
polyno
mials
119867119899(119909
)[11]andfor
119903=
2classical2-orthogon
alpo
lyno
mials)
(XIII)
119892(119905
)=
(1minus
119905 )119898+1
119860(119905
)=
1
(1minus
119905)119898+1
1
(1minus
119905)119898+1119890119909119905
=infin sum119899=0
119866(119898)
119899(119909
)119905119899
TheM
iller-Lee
polyno
mials[1112](for119898
=0the
trun
catedexpo
nentialp
olyn
omials
119890 119899(119909
)[11]andfor
119898=
120573minus
1the
mod
ified
Laguerre
polyno
mials
119891(120573)
119899(119909
)
[13])
(XIV
)119892
(119905)
=(119890119905
+1)
2119905
119860(119905
)=
2119905
(119890119905+
1)
2119905
(119890119905+
1)119890119909119905
=infin sum119899=0
119866119899
(119909)
119905119899 119899
TheG
enocchip
olyn
omials[14
]
4 International Journal of Analysis
The operators and also satisfy the commutationrelation
[ ] = minus = 1 (8)
and thus display the Weyl group structure If the consideredpolynomial set 119901
119899(119909)119899isinN is quasimonomial its properties
can easily be derived from those of the and operatorsIn fact
(i) Combining the recurrences (6) and (7) we have
119901119899
(119909) = 119899119901119899
(119909) (9)
which can be interpreted as the differential equationsatisfied by 119901
119899(119909) if and have a differential
realization(ii) Assuming here and in the sequel 119901
0(119909) = 1 then
119901119899(119909) can be explicitly constructed as
119901119899
(119909) = 119899
1199010
(119909) = 119899
1 (10)
which yields the series definition for 119901119899(119909)
(iii) Identity (10) implies that the exponential generatingfunction of 119901
119899(119909) can be given in the form
exp (119905) 1 =
infin
sum119899=0
119901119899
(119909)119905119899
119899(|119905| lt infin) (11)
We note that the Appell polynomials 119860119899(119909) are quasi-
monomial with respect to the following multiplicative andderivative operators
119860
= 119909 +1198601015840
(119863119909)
119860 (119863119909)
(12a)
or equivalently
119860
= 119909 minus1198921015840
(119863119909)
119892 (119863x) (12b)
119860
= 119863119909 (13)
respectivelyThe special polynomials of two variables are useful from
the point of view of applications in physics Also thesepolynomials allow the derivation of a number of useful iden-tities in a fairly straightforward way and help in introducingnew families of special polynomials For example Bretti etal [17] introduced general classes of two variables Appellpolynomials by using properties of an iterated isomorphismrelated to the Laguerre-type exponentials
We consider the 2-variable general polynomial (2VgP)family 119901
119899(119909 119910) defined by the generating function
119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899(1199010
(119909 119910) = 1) (14)
where 120601(119910 119905) has (at least the formal) series expansion
120601 (119910 119905) =
infin
sum119899=0
120601119899
(119910)119905119899
119899(1206010
(119910) = 0) (15)
We recall that the 2-variable Hermite Kampe de Ferietpolynomials (2VHKdFP) 119867
119899(119909 119910) [18] the Gould-Hopper
polynomials (GHP) 119867(119898)
119899(119909 119910) [19] and the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) [20] are defined by the
generating functions
119890119909119905+119910119905
2
=
infin
sum119899=0
119867119899
(119909 119910)119905119899
119899 (16)
119890119909119905+119910119905
119898
=
infin
sum119899=0
119867(119898)
119899(119909 119910)
119905119899
119899 (17)
119860 (119905) 119890119909119905+119910119905
2
=
infin
sum119899=0
119867119860119899
(119909 119910)119905119899
119899 (18)
respectively Thus in view of generating functions (14) (16)(17) and (18) we note that the 2VHKdFP 119867
119899(119909 119910) the GHP
119867(119898)
119899(119909 119910) and the HAP
119867119860119899(119909 119910) belong to 2VgP family
In this paper operational methods are used to introducecertain new families of special polynomials related to theAppell polynomials In Section 2 some results for the 2-variable general polynomials (2VgP) 119901
119899(119909 119910) are derived
Further the 2-variable general-Appell polynomials (2VgAP)119901119860119899(119909 119910) are introduced and framed within the context
of monomiality principle In Section 3 the Gould-Hopper-Appell polynomials (GHAP)
119867119860(119898)
119899(119909 119910) are considered
and their properties are established Some members belong-ing to theGould-Hopper-Appell polynomial family are given
2 2-Variable General-Appell Polynomials
In order to introduce the 2-variable general-Appell polyno-mials (2VgAP) we need to establish certain results for the2VgP 119901
119899(119909 119910) Therefore first we prove the following results
for the 2VgP 119901119899(119909 119910)
Lemma 1 The 2VgP 119901119899(119909 119910) defined by generating function
(14) where 120601(119910 119905) is given by (15) are quasimonomial underthe action of the following multiplicative and derivative opera-tors
119901
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
(1206011015840
(119909 119905) =120597
120597119905120601 (119909 119905)) (19)
119901
= 119863119909 (20)
respectively
Proof Differentiating (14) partially with respect to 119905 we have
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (21)
International Journal of Analysis 5
Table2Listof
someG
ould-H
opper-A
ppellp
olyn
omials119867
119860(119898)
119899(119909
119910)
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(I)
(119890119905
minus1)
119905
119905
(119890119905minus
1)
119909+
119898119910
119863119898minus1
119909+
((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905
(119890119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Be
rnou
llipo
lyno
mials
(II)
(119890119905
+1)
2
2
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909minus
119890119863119909
(119890119863119909
+1)119863119909
2
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898)
119899(119909
119910)
119905119899 119899
TheG
ould
Hop
per-Eu
ler
polyno
mials
(III)
(119890119905
minus1)120572
119905120572
119905120572
(119890119905minus
1)120572
119909+
119898119910
119863119898minus1
119909+
120572((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905120572
(119890119905minus
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mials
(IV)
(119890119905
+1)120572
2120572
2120572
(119890119905+
1)120572
119909+
119898119910
119863119898minus1
119909minus
120572119890119863119909
(119890119863119909
+1)119863119909
2120572
(119890119905+
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omials
(V)
119904 prod ℎ=1
(119890120572ℎ119905
minus1)
120572ℎ119905119904
119909
+119898
119910119863119898minus1
119909+
119898119863minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909
(119890120572119903119863119909
minus1)119863119909
1205721
sdotsdotsdot120572119904119905119904
(1198901205721119905minus
1)sdotsdot
sdot (119890120572119904119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mialsof
order119904
119904 prod ℎ=1
120572ℎ
(119890120572ℎ119905minus
1)119905119904
(VI)
119904 prod ℎ=1
(119890120572ℎ119905
+1)
2119904
119909
+119898
119910119863119898minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909119863119909
2119904
(1198901205721119905+
1)sdotsdot
sdot (119890120572119904119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omialsof
order119904
119904 prod ℎ=1
2119904
(119890120572ℎ119905+
1)
(VII)
119890119905minus
sum119904minus1
ℎ=0
(119905ℎ
ℎ)
119905119904
119909+
119898119910
119863119898minus1
119909+
119898119863minus1
119909minus
1minus
119863119909
119898minus1
(119898
minus1)
(119890119863119909
minussum119898minus1
ℎ=0
119905ℎℎ
)119863119909
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898[119904minus1])
119899(119909
119910)
119905119899 119899
TheG
ould-H
oppern
ewgeneralized
Bernou
llipo
lyno
mials
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)
6 International Journal of Analysis
Table2Con
tinued
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(VIII)
(119905
120582119890119905
minus1
)minus120572
(119905
120582119890119905
minus1
)120572
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(120572minus
119863119909)
minus120572
119863119909
(120582119890119863119909
minus1)
119863119909
(119905
120582119890119905
minus1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898120572)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mialsof
order120572
(IX)
120582119890119905
minus1
119905
119905
120582119890119905
minus1
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(1minus
119863119909)
minus1
119863119909
(120582119890119863119909
minus1)
119863119909
119905
120582119890119905
minus1
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mials
(X)
(2
120582119890119905
+1
)minus120572
(2
120582119890119905
+1
)120572
119909+
119898119910
119863119898minus1
119909minus
120582120572
119890119863119909
(120582119890119863119909
+1)119863119909
(2
120582119890119905
+1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898120572)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheG
ould-H
opper-A
posto
l-Euler
polyno
mialsof
order120572
(XI)
120582119890119905
+1
2
2
120582119890119905
+1
119909+
119898119910
119863119898minus1
119909minus
120582119890119863119909
(120582119890119863119909
+1)119863119909
2
120582119890119905
+1
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheG
ould-H
opper-A
posto
l-Euler
polyno
mials
(XII)
119890minus(sum119903+1
ℎ=0120585ℎ119905ℎ)
119890(sum119903+1
ℎ=0120585ℎ119905ℎ)
120585 119903+1
=0
119909+
119898119910
119863119898minus1
119909+119903+1
sumℎ=1
ℎ120585 ℎ
119863ℎminus1
119909119863119909
119890(sum119903+1
ℎ=0120585ℎ119905ℎ+119909119905+119910119905119898)
=infin sum119899=0
119867119860(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Gou
ld-H
opperp
olyn
omials(fo
r119903=
1
theG
ould-H
opper-Hermite
polyno
mials119867
119867(119898)
119899(119909
119910)andfor
119903=
2the
Gou
ld-H
opperc
lassical
2-orthogon
alpo
lyno
mials)
(XIII)
(1minus
119905 )119904+1
1
(1minus
119905 )119904+1
119909+
119898119910
119863119898minus1
119909minus
119898+
1
1minus
119863119909
119863119909
1
(1minus
119905 )119904+1119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898119904)
119899(119909
119910)
119905119899
TheG
ould-H
opper-Miller-Lee
polyno
mials(fo
r119904=
0the
Gou
ld-H
opper-trun
catedexpo
nential
polyno
mials119867
119890119898 119899(119909
119910)andfor
119904=
120573minus
1Gou
ld-H
opper-mod
ified
Laguerre
polyno
mials119867
119891(119898120573)
119899(119909
119910))
(XIV
)(119890119905
+1)
2119905
2119905
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909+
119890119863119909
(1minus
119863119909)
+1
(119890119863119909
+1)
119863119909
2119905
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Genocchi
polyno
mials
International Journal of Analysis 7
If 120601(119910 119905) is an invertible series and 1206011015840
(119910 119905)120601(119910 119905) hasTaylorrsquos series expansion in powers of 119905 then in view of theidentity
119863119909
119890119909119905
120601 (119910 119905) = 119905 (119890119909119905
120601 (119910 119905)) (22)
we can write
1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119890119909119905
120601 (119910 119905) =1206011015840
(119910 119905)
120601 (119910 119905)(119890119909119905
120601 (119910 119905)) (23)
Now using (23) in the lhs of (21) we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (24)
Making use of generating function (14) in the lhs of theabove equation we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899
(25)
which on equating the coefficients of like powers of 119905 in bothsides gives
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119901119899
(119909 119910) = 119901119899+1
(119909 119910) (26)
Thus in view of monomiality principle equation (6) theabove equation yields assertion (19) of Lemma 1 Again usingidentity (22) in (14) we have
119863119909
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119899minus1
(119909 119910)119905119899
(119899 minus 1) (27)
Equating the coefficients of like powers of 119905 in both sidesof (27) we find
119863119909
119901119899
(119909 119910) = 119899119901119899minus1
(119909 119910) (119899 ge 1) (28)
which in view of monomiality principle equation (7) yieldsassertion (20) of Lemma 1
Remark 2 The operators given by (19) and (20) satisfycommutation relation (8) Also using expressions (19) and(20) in (9) we get the following differential equation satisfiedby 2VgP 119901
119899(119909 119910)
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus 119899) 119901119899
(119909 119910) = 0 (29)
Remark 3 Since 1199010(119909 119910) = 1 therefore in view of monomi-
ality principle equation (10) we have
119901119899
(119909 119910) = (119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
119899
1 (1199010
(119909 119910) = 1) (30)
Also in view of (11) (14) and (19) we have
exp (119901119905) 1 = 119890
119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 (31)
Now we proceed to introduce the 2-variable general-Appell polynomials (2VgAP) In order to derive the gener-ating functions for the 2VgAP we take the 2VgP 119901
119899(119909 119910) as
the base in the Appell polynomials generating function (1)Thus replacing 119909 by the multiplicative operator
119901of the
2VgP 119901119899(119909 119910) in the lhs of (1) and denoting the resultant
2VgAP by119901119860119899(119909 119910) we have
119860 (119905) 119890(119901119905)
=
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (32)
Now using (31) in the exponential term in the lhs of(32) we get the generating function for
119901119860119899(119909 119910) as
119860 (119905) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (33)
In view of (5) generating function (33) can be expressedequivalently as
1
119892 (119905)119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (34)
Now we frame the 2VgAP119901119860119899(119909 119910) within the context
of monomiality principle formalism We prove the followingresults
Theorem 4 The 2VgAP119901119860119899(119909 119910) are quasimonomial with
respect to the following multiplicative and derivative operators
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
(35a)
or equivalently
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
minus1198921015840
(119863119909)
119892 (119863119909)
(35b)
119901119860
= 119863119909 (36)
respectively
Proof Differentiating (33) partially with respect to 119905 we find
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)+
1198601015840
(119905)
119860 (119905)) 119860 (119905) 119890
119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(37)
Since 119860(119905) and 120601(119910 119905) are invertible series of 119905 therefore1198601015840
(119905)119860(119905) and 1206011015840
(119910 119905)120601(119910 119905) possess power series expan-sions of 119905 Thus in view of identity (22) we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119860 (119905) 119890119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(38)
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 3
Table1Con
tinued
SNo
119892(119905
)119860
(119905)
Generatingfunctio
nsPo
lyno
mials
(VIII)
119892(119905
)=
(119905
120582119890119905
minus1
)minus120572
119860(119905
)=
(119905
120582119890119905
minus1
)120572
(119905
120582119890119905
minus1
)120572
119890119909119905
=infin sum119899=0
B(120572)
119899(119909
120582)
119905119899 119899
TheA
posto
l-Berno
ullipo
lyno
mialsof
order120572
[7]
(IX)
119892(119905
)=
120582119890119905
minus1
119905119860
(119905)
=119905
120582119890119905
minus1
119905
120582119890119905
minus1
119890119909119905
=infin sum119899=0
B119899
(119909120582
)119905119899 119899
TheA
posto
l-Berno
ullipo
lyno
mials[78]
(X)
119892(119905
)=
(2
120582119890119905
+1
)minus120572
119860(119905
)=
(2
120582119890119905
+1
)120572
(2
120582119890119905
+1
)120572
119890119909119905
=infin sum119899=0
E(120572)
119899(119909
120582)
119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheA
posto
l-Euler
polyno
mialsof
order120572
[79]
(XI)
119892(119905
)=
120582119890119905
+1
2119860
(119905)
=2
120582119890119905
+1
2
120582119890119905
+1
119890119909119905
=infin sum119899=0
E119899
(119909120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheA
posto
l-Euler
polyno
mials[7ndash9
]
(XII)
119892(119905
)=
119890minus(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
119860
(119905)
=119890(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
120585 119903+1
=0
119890(120585∘+1205851119905+12058521199052+sdotsdotsdot+120585119903+1119905119903+1)
119890119909119905
=infin sum119899=0
119860119899
(119909)
119905119899 119899
Theg
eneralized
Gou
ld-H
opperp
olyn
omials[10]
(for
119903=
1the
Hermite
polyno
mials
119867119899(119909
)[11]andfor
119903=
2classical2-orthogon
alpo
lyno
mials)
(XIII)
119892(119905
)=
(1minus
119905 )119898+1
119860(119905
)=
1
(1minus
119905)119898+1
1
(1minus
119905)119898+1119890119909119905
=infin sum119899=0
119866(119898)
119899(119909
)119905119899
TheM
iller-Lee
polyno
mials[1112](for119898
=0the
trun
catedexpo
nentialp
olyn
omials
119890 119899(119909
)[11]andfor
119898=
120573minus
1the
mod
ified
Laguerre
polyno
mials
119891(120573)
119899(119909
)
[13])
(XIV
)119892
(119905)
=(119890119905
+1)
2119905
119860(119905
)=
2119905
(119890119905+
1)
2119905
(119890119905+
1)119890119909119905
=infin sum119899=0
119866119899
(119909)
119905119899 119899
TheG
enocchip
olyn
omials[14
]
4 International Journal of Analysis
The operators and also satisfy the commutationrelation
[ ] = minus = 1 (8)
and thus display the Weyl group structure If the consideredpolynomial set 119901
119899(119909)119899isinN is quasimonomial its properties
can easily be derived from those of the and operatorsIn fact
(i) Combining the recurrences (6) and (7) we have
119901119899
(119909) = 119899119901119899
(119909) (9)
which can be interpreted as the differential equationsatisfied by 119901
119899(119909) if and have a differential
realization(ii) Assuming here and in the sequel 119901
0(119909) = 1 then
119901119899(119909) can be explicitly constructed as
119901119899
(119909) = 119899
1199010
(119909) = 119899
1 (10)
which yields the series definition for 119901119899(119909)
(iii) Identity (10) implies that the exponential generatingfunction of 119901
119899(119909) can be given in the form
exp (119905) 1 =
infin
sum119899=0
119901119899
(119909)119905119899
119899(|119905| lt infin) (11)
We note that the Appell polynomials 119860119899(119909) are quasi-
monomial with respect to the following multiplicative andderivative operators
119860
= 119909 +1198601015840
(119863119909)
119860 (119863119909)
(12a)
or equivalently
119860
= 119909 minus1198921015840
(119863119909)
119892 (119863x) (12b)
119860
= 119863119909 (13)
respectivelyThe special polynomials of two variables are useful from
the point of view of applications in physics Also thesepolynomials allow the derivation of a number of useful iden-tities in a fairly straightforward way and help in introducingnew families of special polynomials For example Bretti etal [17] introduced general classes of two variables Appellpolynomials by using properties of an iterated isomorphismrelated to the Laguerre-type exponentials
We consider the 2-variable general polynomial (2VgP)family 119901
119899(119909 119910) defined by the generating function
119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899(1199010
(119909 119910) = 1) (14)
where 120601(119910 119905) has (at least the formal) series expansion
120601 (119910 119905) =
infin
sum119899=0
120601119899
(119910)119905119899
119899(1206010
(119910) = 0) (15)
We recall that the 2-variable Hermite Kampe de Ferietpolynomials (2VHKdFP) 119867
119899(119909 119910) [18] the Gould-Hopper
polynomials (GHP) 119867(119898)
119899(119909 119910) [19] and the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) [20] are defined by the
generating functions
119890119909119905+119910119905
2
=
infin
sum119899=0
119867119899
(119909 119910)119905119899
119899 (16)
119890119909119905+119910119905
119898
=
infin
sum119899=0
119867(119898)
119899(119909 119910)
119905119899
119899 (17)
119860 (119905) 119890119909119905+119910119905
2
=
infin
sum119899=0
119867119860119899
(119909 119910)119905119899
119899 (18)
respectively Thus in view of generating functions (14) (16)(17) and (18) we note that the 2VHKdFP 119867
119899(119909 119910) the GHP
119867(119898)
119899(119909 119910) and the HAP
119867119860119899(119909 119910) belong to 2VgP family
In this paper operational methods are used to introducecertain new families of special polynomials related to theAppell polynomials In Section 2 some results for the 2-variable general polynomials (2VgP) 119901
119899(119909 119910) are derived
Further the 2-variable general-Appell polynomials (2VgAP)119901119860119899(119909 119910) are introduced and framed within the context
of monomiality principle In Section 3 the Gould-Hopper-Appell polynomials (GHAP)
119867119860(119898)
119899(119909 119910) are considered
and their properties are established Some members belong-ing to theGould-Hopper-Appell polynomial family are given
2 2-Variable General-Appell Polynomials
In order to introduce the 2-variable general-Appell polyno-mials (2VgAP) we need to establish certain results for the2VgP 119901
119899(119909 119910) Therefore first we prove the following results
for the 2VgP 119901119899(119909 119910)
Lemma 1 The 2VgP 119901119899(119909 119910) defined by generating function
(14) where 120601(119910 119905) is given by (15) are quasimonomial underthe action of the following multiplicative and derivative opera-tors
119901
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
(1206011015840
(119909 119905) =120597
120597119905120601 (119909 119905)) (19)
119901
= 119863119909 (20)
respectively
Proof Differentiating (14) partially with respect to 119905 we have
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (21)
International Journal of Analysis 5
Table2Listof
someG
ould-H
opper-A
ppellp
olyn
omials119867
119860(119898)
119899(119909
119910)
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(I)
(119890119905
minus1)
119905
119905
(119890119905minus
1)
119909+
119898119910
119863119898minus1
119909+
((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905
(119890119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Be
rnou
llipo
lyno
mials
(II)
(119890119905
+1)
2
2
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909minus
119890119863119909
(119890119863119909
+1)119863119909
2
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898)
119899(119909
119910)
119905119899 119899
TheG
ould
Hop
per-Eu
ler
polyno
mials
(III)
(119890119905
minus1)120572
119905120572
119905120572
(119890119905minus
1)120572
119909+
119898119910
119863119898minus1
119909+
120572((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905120572
(119890119905minus
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mials
(IV)
(119890119905
+1)120572
2120572
2120572
(119890119905+
1)120572
119909+
119898119910
119863119898minus1
119909minus
120572119890119863119909
(119890119863119909
+1)119863119909
2120572
(119890119905+
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omials
(V)
119904 prod ℎ=1
(119890120572ℎ119905
minus1)
120572ℎ119905119904
119909
+119898
119910119863119898minus1
119909+
119898119863minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909
(119890120572119903119863119909
minus1)119863119909
1205721
sdotsdotsdot120572119904119905119904
(1198901205721119905minus
1)sdotsdot
sdot (119890120572119904119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mialsof
order119904
119904 prod ℎ=1
120572ℎ
(119890120572ℎ119905minus
1)119905119904
(VI)
119904 prod ℎ=1
(119890120572ℎ119905
+1)
2119904
119909
+119898
119910119863119898minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909119863119909
2119904
(1198901205721119905+
1)sdotsdot
sdot (119890120572119904119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omialsof
order119904
119904 prod ℎ=1
2119904
(119890120572ℎ119905+
1)
(VII)
119890119905minus
sum119904minus1
ℎ=0
(119905ℎ
ℎ)
119905119904
119909+
119898119910
119863119898minus1
119909+
119898119863minus1
119909minus
1minus
119863119909
119898minus1
(119898
minus1)
(119890119863119909
minussum119898minus1
ℎ=0
119905ℎℎ
)119863119909
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898[119904minus1])
119899(119909
119910)
119905119899 119899
TheG
ould-H
oppern
ewgeneralized
Bernou
llipo
lyno
mials
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)
6 International Journal of Analysis
Table2Con
tinued
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(VIII)
(119905
120582119890119905
minus1
)minus120572
(119905
120582119890119905
minus1
)120572
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(120572minus
119863119909)
minus120572
119863119909
(120582119890119863119909
minus1)
119863119909
(119905
120582119890119905
minus1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898120572)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mialsof
order120572
(IX)
120582119890119905
minus1
119905
119905
120582119890119905
minus1
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(1minus
119863119909)
minus1
119863119909
(120582119890119863119909
minus1)
119863119909
119905
120582119890119905
minus1
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mials
(X)
(2
120582119890119905
+1
)minus120572
(2
120582119890119905
+1
)120572
119909+
119898119910
119863119898minus1
119909minus
120582120572
119890119863119909
(120582119890119863119909
+1)119863119909
(2
120582119890119905
+1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898120572)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheG
ould-H
opper-A
posto
l-Euler
polyno
mialsof
order120572
(XI)
120582119890119905
+1
2
2
120582119890119905
+1
119909+
119898119910
119863119898minus1
119909minus
120582119890119863119909
(120582119890119863119909
+1)119863119909
2
120582119890119905
+1
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheG
ould-H
opper-A
posto
l-Euler
polyno
mials
(XII)
119890minus(sum119903+1
ℎ=0120585ℎ119905ℎ)
119890(sum119903+1
ℎ=0120585ℎ119905ℎ)
120585 119903+1
=0
119909+
119898119910
119863119898minus1
119909+119903+1
sumℎ=1
ℎ120585 ℎ
119863ℎminus1
119909119863119909
119890(sum119903+1
ℎ=0120585ℎ119905ℎ+119909119905+119910119905119898)
=infin sum119899=0
119867119860(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Gou
ld-H
opperp
olyn
omials(fo
r119903=
1
theG
ould-H
opper-Hermite
polyno
mials119867
119867(119898)
119899(119909
119910)andfor
119903=
2the
Gou
ld-H
opperc
lassical
2-orthogon
alpo
lyno
mials)
(XIII)
(1minus
119905 )119904+1
1
(1minus
119905 )119904+1
119909+
119898119910
119863119898minus1
119909minus
119898+
1
1minus
119863119909
119863119909
1
(1minus
119905 )119904+1119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898119904)
119899(119909
119910)
119905119899
TheG
ould-H
opper-Miller-Lee
polyno
mials(fo
r119904=
0the
Gou
ld-H
opper-trun
catedexpo
nential
polyno
mials119867
119890119898 119899(119909
119910)andfor
119904=
120573minus
1Gou
ld-H
opper-mod
ified
Laguerre
polyno
mials119867
119891(119898120573)
119899(119909
119910))
(XIV
)(119890119905
+1)
2119905
2119905
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909+
119890119863119909
(1minus
119863119909)
+1
(119890119863119909
+1)
119863119909
2119905
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Genocchi
polyno
mials
International Journal of Analysis 7
If 120601(119910 119905) is an invertible series and 1206011015840
(119910 119905)120601(119910 119905) hasTaylorrsquos series expansion in powers of 119905 then in view of theidentity
119863119909
119890119909119905
120601 (119910 119905) = 119905 (119890119909119905
120601 (119910 119905)) (22)
we can write
1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119890119909119905
120601 (119910 119905) =1206011015840
(119910 119905)
120601 (119910 119905)(119890119909119905
120601 (119910 119905)) (23)
Now using (23) in the lhs of (21) we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (24)
Making use of generating function (14) in the lhs of theabove equation we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899
(25)
which on equating the coefficients of like powers of 119905 in bothsides gives
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119901119899
(119909 119910) = 119901119899+1
(119909 119910) (26)
Thus in view of monomiality principle equation (6) theabove equation yields assertion (19) of Lemma 1 Again usingidentity (22) in (14) we have
119863119909
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119899minus1
(119909 119910)119905119899
(119899 minus 1) (27)
Equating the coefficients of like powers of 119905 in both sidesof (27) we find
119863119909
119901119899
(119909 119910) = 119899119901119899minus1
(119909 119910) (119899 ge 1) (28)
which in view of monomiality principle equation (7) yieldsassertion (20) of Lemma 1
Remark 2 The operators given by (19) and (20) satisfycommutation relation (8) Also using expressions (19) and(20) in (9) we get the following differential equation satisfiedby 2VgP 119901
119899(119909 119910)
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus 119899) 119901119899
(119909 119910) = 0 (29)
Remark 3 Since 1199010(119909 119910) = 1 therefore in view of monomi-
ality principle equation (10) we have
119901119899
(119909 119910) = (119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
119899
1 (1199010
(119909 119910) = 1) (30)
Also in view of (11) (14) and (19) we have
exp (119901119905) 1 = 119890
119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 (31)
Now we proceed to introduce the 2-variable general-Appell polynomials (2VgAP) In order to derive the gener-ating functions for the 2VgAP we take the 2VgP 119901
119899(119909 119910) as
the base in the Appell polynomials generating function (1)Thus replacing 119909 by the multiplicative operator
119901of the
2VgP 119901119899(119909 119910) in the lhs of (1) and denoting the resultant
2VgAP by119901119860119899(119909 119910) we have
119860 (119905) 119890(119901119905)
=
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (32)
Now using (31) in the exponential term in the lhs of(32) we get the generating function for
119901119860119899(119909 119910) as
119860 (119905) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (33)
In view of (5) generating function (33) can be expressedequivalently as
1
119892 (119905)119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (34)
Now we frame the 2VgAP119901119860119899(119909 119910) within the context
of monomiality principle formalism We prove the followingresults
Theorem 4 The 2VgAP119901119860119899(119909 119910) are quasimonomial with
respect to the following multiplicative and derivative operators
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
(35a)
or equivalently
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
minus1198921015840
(119863119909)
119892 (119863119909)
(35b)
119901119860
= 119863119909 (36)
respectively
Proof Differentiating (33) partially with respect to 119905 we find
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)+
1198601015840
(119905)
119860 (119905)) 119860 (119905) 119890
119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(37)
Since 119860(119905) and 120601(119910 119905) are invertible series of 119905 therefore1198601015840
(119905)119860(119905) and 1206011015840
(119910 119905)120601(119910 119905) possess power series expan-sions of 119905 Thus in view of identity (22) we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119860 (119905) 119890119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(38)
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 International Journal of Analysis
The operators and also satisfy the commutationrelation
[ ] = minus = 1 (8)
and thus display the Weyl group structure If the consideredpolynomial set 119901
119899(119909)119899isinN is quasimonomial its properties
can easily be derived from those of the and operatorsIn fact
(i) Combining the recurrences (6) and (7) we have
119901119899
(119909) = 119899119901119899
(119909) (9)
which can be interpreted as the differential equationsatisfied by 119901
119899(119909) if and have a differential
realization(ii) Assuming here and in the sequel 119901
0(119909) = 1 then
119901119899(119909) can be explicitly constructed as
119901119899
(119909) = 119899
1199010
(119909) = 119899
1 (10)
which yields the series definition for 119901119899(119909)
(iii) Identity (10) implies that the exponential generatingfunction of 119901
119899(119909) can be given in the form
exp (119905) 1 =
infin
sum119899=0
119901119899
(119909)119905119899
119899(|119905| lt infin) (11)
We note that the Appell polynomials 119860119899(119909) are quasi-
monomial with respect to the following multiplicative andderivative operators
119860
= 119909 +1198601015840
(119863119909)
119860 (119863119909)
(12a)
or equivalently
119860
= 119909 minus1198921015840
(119863119909)
119892 (119863x) (12b)
119860
= 119863119909 (13)
respectivelyThe special polynomials of two variables are useful from
the point of view of applications in physics Also thesepolynomials allow the derivation of a number of useful iden-tities in a fairly straightforward way and help in introducingnew families of special polynomials For example Bretti etal [17] introduced general classes of two variables Appellpolynomials by using properties of an iterated isomorphismrelated to the Laguerre-type exponentials
We consider the 2-variable general polynomial (2VgP)family 119901
119899(119909 119910) defined by the generating function
119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899(1199010
(119909 119910) = 1) (14)
where 120601(119910 119905) has (at least the formal) series expansion
120601 (119910 119905) =
infin
sum119899=0
120601119899
(119910)119905119899
119899(1206010
(119910) = 0) (15)
We recall that the 2-variable Hermite Kampe de Ferietpolynomials (2VHKdFP) 119867
119899(119909 119910) [18] the Gould-Hopper
polynomials (GHP) 119867(119898)
119899(119909 119910) [19] and the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) [20] are defined by the
generating functions
119890119909119905+119910119905
2
=
infin
sum119899=0
119867119899
(119909 119910)119905119899
119899 (16)
119890119909119905+119910119905
119898
=
infin
sum119899=0
119867(119898)
119899(119909 119910)
119905119899
119899 (17)
119860 (119905) 119890119909119905+119910119905
2
=
infin
sum119899=0
119867119860119899
(119909 119910)119905119899
119899 (18)
respectively Thus in view of generating functions (14) (16)(17) and (18) we note that the 2VHKdFP 119867
119899(119909 119910) the GHP
119867(119898)
119899(119909 119910) and the HAP
119867119860119899(119909 119910) belong to 2VgP family
In this paper operational methods are used to introducecertain new families of special polynomials related to theAppell polynomials In Section 2 some results for the 2-variable general polynomials (2VgP) 119901
119899(119909 119910) are derived
Further the 2-variable general-Appell polynomials (2VgAP)119901119860119899(119909 119910) are introduced and framed within the context
of monomiality principle In Section 3 the Gould-Hopper-Appell polynomials (GHAP)
119867119860(119898)
119899(119909 119910) are considered
and their properties are established Some members belong-ing to theGould-Hopper-Appell polynomial family are given
2 2-Variable General-Appell Polynomials
In order to introduce the 2-variable general-Appell polyno-mials (2VgAP) we need to establish certain results for the2VgP 119901
119899(119909 119910) Therefore first we prove the following results
for the 2VgP 119901119899(119909 119910)
Lemma 1 The 2VgP 119901119899(119909 119910) defined by generating function
(14) where 120601(119910 119905) is given by (15) are quasimonomial underthe action of the following multiplicative and derivative opera-tors
119901
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
(1206011015840
(119909 119905) =120597
120597119905120601 (119909 119905)) (19)
119901
= 119863119909 (20)
respectively
Proof Differentiating (14) partially with respect to 119905 we have
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (21)
International Journal of Analysis 5
Table2Listof
someG
ould-H
opper-A
ppellp
olyn
omials119867
119860(119898)
119899(119909
119910)
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(I)
(119890119905
minus1)
119905
119905
(119890119905minus
1)
119909+
119898119910
119863119898minus1
119909+
((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905
(119890119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Be
rnou
llipo
lyno
mials
(II)
(119890119905
+1)
2
2
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909minus
119890119863119909
(119890119863119909
+1)119863119909
2
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898)
119899(119909
119910)
119905119899 119899
TheG
ould
Hop
per-Eu
ler
polyno
mials
(III)
(119890119905
minus1)120572
119905120572
119905120572
(119890119905minus
1)120572
119909+
119898119910
119863119898minus1
119909+
120572((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905120572
(119890119905minus
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mials
(IV)
(119890119905
+1)120572
2120572
2120572
(119890119905+
1)120572
119909+
119898119910
119863119898minus1
119909minus
120572119890119863119909
(119890119863119909
+1)119863119909
2120572
(119890119905+
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omials
(V)
119904 prod ℎ=1
(119890120572ℎ119905
minus1)
120572ℎ119905119904
119909
+119898
119910119863119898minus1
119909+
119898119863minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909
(119890120572119903119863119909
minus1)119863119909
1205721
sdotsdotsdot120572119904119905119904
(1198901205721119905minus
1)sdotsdot
sdot (119890120572119904119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mialsof
order119904
119904 prod ℎ=1
120572ℎ
(119890120572ℎ119905minus
1)119905119904
(VI)
119904 prod ℎ=1
(119890120572ℎ119905
+1)
2119904
119909
+119898
119910119863119898minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909119863119909
2119904
(1198901205721119905+
1)sdotsdot
sdot (119890120572119904119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omialsof
order119904
119904 prod ℎ=1
2119904
(119890120572ℎ119905+
1)
(VII)
119890119905minus
sum119904minus1
ℎ=0
(119905ℎ
ℎ)
119905119904
119909+
119898119910
119863119898minus1
119909+
119898119863minus1
119909minus
1minus
119863119909
119898minus1
(119898
minus1)
(119890119863119909
minussum119898minus1
ℎ=0
119905ℎℎ
)119863119909
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898[119904minus1])
119899(119909
119910)
119905119899 119899
TheG
ould-H
oppern
ewgeneralized
Bernou
llipo
lyno
mials
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)
6 International Journal of Analysis
Table2Con
tinued
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(VIII)
(119905
120582119890119905
minus1
)minus120572
(119905
120582119890119905
minus1
)120572
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(120572minus
119863119909)
minus120572
119863119909
(120582119890119863119909
minus1)
119863119909
(119905
120582119890119905
minus1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898120572)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mialsof
order120572
(IX)
120582119890119905
minus1
119905
119905
120582119890119905
minus1
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(1minus
119863119909)
minus1
119863119909
(120582119890119863119909
minus1)
119863119909
119905
120582119890119905
minus1
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mials
(X)
(2
120582119890119905
+1
)minus120572
(2
120582119890119905
+1
)120572
119909+
119898119910
119863119898minus1
119909minus
120582120572
119890119863119909
(120582119890119863119909
+1)119863119909
(2
120582119890119905
+1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898120572)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheG
ould-H
opper-A
posto
l-Euler
polyno
mialsof
order120572
(XI)
120582119890119905
+1
2
2
120582119890119905
+1
119909+
119898119910
119863119898minus1
119909minus
120582119890119863119909
(120582119890119863119909
+1)119863119909
2
120582119890119905
+1
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheG
ould-H
opper-A
posto
l-Euler
polyno
mials
(XII)
119890minus(sum119903+1
ℎ=0120585ℎ119905ℎ)
119890(sum119903+1
ℎ=0120585ℎ119905ℎ)
120585 119903+1
=0
119909+
119898119910
119863119898minus1
119909+119903+1
sumℎ=1
ℎ120585 ℎ
119863ℎminus1
119909119863119909
119890(sum119903+1
ℎ=0120585ℎ119905ℎ+119909119905+119910119905119898)
=infin sum119899=0
119867119860(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Gou
ld-H
opperp
olyn
omials(fo
r119903=
1
theG
ould-H
opper-Hermite
polyno
mials119867
119867(119898)
119899(119909
119910)andfor
119903=
2the
Gou
ld-H
opperc
lassical
2-orthogon
alpo
lyno
mials)
(XIII)
(1minus
119905 )119904+1
1
(1minus
119905 )119904+1
119909+
119898119910
119863119898minus1
119909minus
119898+
1
1minus
119863119909
119863119909
1
(1minus
119905 )119904+1119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898119904)
119899(119909
119910)
119905119899
TheG
ould-H
opper-Miller-Lee
polyno
mials(fo
r119904=
0the
Gou
ld-H
opper-trun
catedexpo
nential
polyno
mials119867
119890119898 119899(119909
119910)andfor
119904=
120573minus
1Gou
ld-H
opper-mod
ified
Laguerre
polyno
mials119867
119891(119898120573)
119899(119909
119910))
(XIV
)(119890119905
+1)
2119905
2119905
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909+
119890119863119909
(1minus
119863119909)
+1
(119890119863119909
+1)
119863119909
2119905
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Genocchi
polyno
mials
International Journal of Analysis 7
If 120601(119910 119905) is an invertible series and 1206011015840
(119910 119905)120601(119910 119905) hasTaylorrsquos series expansion in powers of 119905 then in view of theidentity
119863119909
119890119909119905
120601 (119910 119905) = 119905 (119890119909119905
120601 (119910 119905)) (22)
we can write
1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119890119909119905
120601 (119910 119905) =1206011015840
(119910 119905)
120601 (119910 119905)(119890119909119905
120601 (119910 119905)) (23)
Now using (23) in the lhs of (21) we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (24)
Making use of generating function (14) in the lhs of theabove equation we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899
(25)
which on equating the coefficients of like powers of 119905 in bothsides gives
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119901119899
(119909 119910) = 119901119899+1
(119909 119910) (26)
Thus in view of monomiality principle equation (6) theabove equation yields assertion (19) of Lemma 1 Again usingidentity (22) in (14) we have
119863119909
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119899minus1
(119909 119910)119905119899
(119899 minus 1) (27)
Equating the coefficients of like powers of 119905 in both sidesof (27) we find
119863119909
119901119899
(119909 119910) = 119899119901119899minus1
(119909 119910) (119899 ge 1) (28)
which in view of monomiality principle equation (7) yieldsassertion (20) of Lemma 1
Remark 2 The operators given by (19) and (20) satisfycommutation relation (8) Also using expressions (19) and(20) in (9) we get the following differential equation satisfiedby 2VgP 119901
119899(119909 119910)
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus 119899) 119901119899
(119909 119910) = 0 (29)
Remark 3 Since 1199010(119909 119910) = 1 therefore in view of monomi-
ality principle equation (10) we have
119901119899
(119909 119910) = (119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
119899
1 (1199010
(119909 119910) = 1) (30)
Also in view of (11) (14) and (19) we have
exp (119901119905) 1 = 119890
119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 (31)
Now we proceed to introduce the 2-variable general-Appell polynomials (2VgAP) In order to derive the gener-ating functions for the 2VgAP we take the 2VgP 119901
119899(119909 119910) as
the base in the Appell polynomials generating function (1)Thus replacing 119909 by the multiplicative operator
119901of the
2VgP 119901119899(119909 119910) in the lhs of (1) and denoting the resultant
2VgAP by119901119860119899(119909 119910) we have
119860 (119905) 119890(119901119905)
=
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (32)
Now using (31) in the exponential term in the lhs of(32) we get the generating function for
119901119860119899(119909 119910) as
119860 (119905) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (33)
In view of (5) generating function (33) can be expressedequivalently as
1
119892 (119905)119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (34)
Now we frame the 2VgAP119901119860119899(119909 119910) within the context
of monomiality principle formalism We prove the followingresults
Theorem 4 The 2VgAP119901119860119899(119909 119910) are quasimonomial with
respect to the following multiplicative and derivative operators
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
(35a)
or equivalently
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
minus1198921015840
(119863119909)
119892 (119863119909)
(35b)
119901119860
= 119863119909 (36)
respectively
Proof Differentiating (33) partially with respect to 119905 we find
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)+
1198601015840
(119905)
119860 (119905)) 119860 (119905) 119890
119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(37)
Since 119860(119905) and 120601(119910 119905) are invertible series of 119905 therefore1198601015840
(119905)119860(119905) and 1206011015840
(119910 119905)120601(119910 119905) possess power series expan-sions of 119905 Thus in view of identity (22) we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119860 (119905) 119890119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(38)
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 5
Table2Listof
someG
ould-H
opper-A
ppellp
olyn
omials119867
119860(119898)
119899(119909
119910)
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(I)
(119890119905
minus1)
119905
119905
(119890119905minus
1)
119909+
119898119910
119863119898minus1
119909+
((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905
(119890119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Be
rnou
llipo
lyno
mials
(II)
(119890119905
+1)
2
2
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909minus
119890119863119909
(119890119863119909
+1)119863119909
2
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898)
119899(119909
119910)
119905119899 119899
TheG
ould
Hop
per-Eu
ler
polyno
mials
(III)
(119890119905
minus1)120572
119905120572
119905120572
(119890119905minus
1)120572
119909+
119898119910
119863119898minus1
119909+
120572((1
minus119863119909)
119890119863119909
minus1)
119863119909
(119890119863119909
minus1)
119863119909
119905120572
(119890119905minus
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mials
(IV)
(119890119905
+1)120572
2120572
2120572
(119890119905+
1)120572
119909+
119898119910
119863119898minus1
119909minus
120572119890119863119909
(119890119863119909
+1)119863119909
2120572
(119890119905+
1)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898120572)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omials
(V)
119904 prod ℎ=1
(119890120572ℎ119905
minus1)
120572ℎ119905119904
119909
+119898
119910119863119898minus1
119909+
119898119863minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909
(119890120572119903119863119909
minus1)119863119909
1205721
sdotsdotsdot120572119904119905119904
(1198901205721119905minus
1)sdotsdot
sdot (119890120572119904119905minus
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Bernou
llipo
lyno
mialsof
order119904
119904 prod ℎ=1
120572ℎ
(119890120572ℎ119905minus
1)119905119904
(VI)
119904 prod ℎ=1
(119890120572ℎ119905
+1)
2119904
119909
+119898
119910119863119898minus1
119909minus119898 sum 119903=1
120572119903119890120572119903119863119909119863119909
2119904
(1198901205721119905+
1)sdotsdot
sdot (119890120572119904119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119864(119898119904)
119899(119909
119910|
1205721
120572119904)119905119899 119899
TheG
ould-H
opper-generalized
Eulerp
olyn
omialsof
order119904
119904 prod ℎ=1
2119904
(119890120572ℎ119905+
1)
(VII)
119890119905minus
sum119904minus1
ℎ=0
(119905ℎ
ℎ)
119905119904
119909+
119898119910
119863119898minus1
119909+
119898119863minus1
119909minus
1minus
119863119909
119898minus1
(119898
minus1)
(119890119863119909
minussum119898minus1
ℎ=0
119905ℎℎ
)119863119909
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)119890119909119905+119910119905119898
=infin sum119899=0
119867119861(119898[119904minus1])
119899(119909
119910)
119905119899 119899
TheG
ould-H
oppern
ewgeneralized
Bernou
llipo
lyno
mials
119905119904
119890119905minus
sum119904minus1
ℎ=0
(119905ℎℎ
)
6 International Journal of Analysis
Table2Con
tinued
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(VIII)
(119905
120582119890119905
minus1
)minus120572
(119905
120582119890119905
minus1
)120572
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(120572minus
119863119909)
minus120572
119863119909
(120582119890119863119909
minus1)
119863119909
(119905
120582119890119905
minus1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898120572)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mialsof
order120572
(IX)
120582119890119905
minus1
119905
119905
120582119890119905
minus1
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(1minus
119863119909)
minus1
119863119909
(120582119890119863119909
minus1)
119863119909
119905
120582119890119905
minus1
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mials
(X)
(2
120582119890119905
+1
)minus120572
(2
120582119890119905
+1
)120572
119909+
119898119910
119863119898minus1
119909minus
120582120572
119890119863119909
(120582119890119863119909
+1)119863119909
(2
120582119890119905
+1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898120572)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheG
ould-H
opper-A
posto
l-Euler
polyno
mialsof
order120572
(XI)
120582119890119905
+1
2
2
120582119890119905
+1
119909+
119898119910
119863119898minus1
119909minus
120582119890119863119909
(120582119890119863119909
+1)119863119909
2
120582119890119905
+1
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheG
ould-H
opper-A
posto
l-Euler
polyno
mials
(XII)
119890minus(sum119903+1
ℎ=0120585ℎ119905ℎ)
119890(sum119903+1
ℎ=0120585ℎ119905ℎ)
120585 119903+1
=0
119909+
119898119910
119863119898minus1
119909+119903+1
sumℎ=1
ℎ120585 ℎ
119863ℎminus1
119909119863119909
119890(sum119903+1
ℎ=0120585ℎ119905ℎ+119909119905+119910119905119898)
=infin sum119899=0
119867119860(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Gou
ld-H
opperp
olyn
omials(fo
r119903=
1
theG
ould-H
opper-Hermite
polyno
mials119867
119867(119898)
119899(119909
119910)andfor
119903=
2the
Gou
ld-H
opperc
lassical
2-orthogon
alpo
lyno
mials)
(XIII)
(1minus
119905 )119904+1
1
(1minus
119905 )119904+1
119909+
119898119910
119863119898minus1
119909minus
119898+
1
1minus
119863119909
119863119909
1
(1minus
119905 )119904+1119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898119904)
119899(119909
119910)
119905119899
TheG
ould-H
opper-Miller-Lee
polyno
mials(fo
r119904=
0the
Gou
ld-H
opper-trun
catedexpo
nential
polyno
mials119867
119890119898 119899(119909
119910)andfor
119904=
120573minus
1Gou
ld-H
opper-mod
ified
Laguerre
polyno
mials119867
119891(119898120573)
119899(119909
119910))
(XIV
)(119890119905
+1)
2119905
2119905
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909+
119890119863119909
(1minus
119863119909)
+1
(119890119863119909
+1)
119863119909
2119905
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Genocchi
polyno
mials
International Journal of Analysis 7
If 120601(119910 119905) is an invertible series and 1206011015840
(119910 119905)120601(119910 119905) hasTaylorrsquos series expansion in powers of 119905 then in view of theidentity
119863119909
119890119909119905
120601 (119910 119905) = 119905 (119890119909119905
120601 (119910 119905)) (22)
we can write
1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119890119909119905
120601 (119910 119905) =1206011015840
(119910 119905)
120601 (119910 119905)(119890119909119905
120601 (119910 119905)) (23)
Now using (23) in the lhs of (21) we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (24)
Making use of generating function (14) in the lhs of theabove equation we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899
(25)
which on equating the coefficients of like powers of 119905 in bothsides gives
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119901119899
(119909 119910) = 119901119899+1
(119909 119910) (26)
Thus in view of monomiality principle equation (6) theabove equation yields assertion (19) of Lemma 1 Again usingidentity (22) in (14) we have
119863119909
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119899minus1
(119909 119910)119905119899
(119899 minus 1) (27)
Equating the coefficients of like powers of 119905 in both sidesof (27) we find
119863119909
119901119899
(119909 119910) = 119899119901119899minus1
(119909 119910) (119899 ge 1) (28)
which in view of monomiality principle equation (7) yieldsassertion (20) of Lemma 1
Remark 2 The operators given by (19) and (20) satisfycommutation relation (8) Also using expressions (19) and(20) in (9) we get the following differential equation satisfiedby 2VgP 119901
119899(119909 119910)
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus 119899) 119901119899
(119909 119910) = 0 (29)
Remark 3 Since 1199010(119909 119910) = 1 therefore in view of monomi-
ality principle equation (10) we have
119901119899
(119909 119910) = (119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
119899
1 (1199010
(119909 119910) = 1) (30)
Also in view of (11) (14) and (19) we have
exp (119901119905) 1 = 119890
119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 (31)
Now we proceed to introduce the 2-variable general-Appell polynomials (2VgAP) In order to derive the gener-ating functions for the 2VgAP we take the 2VgP 119901
119899(119909 119910) as
the base in the Appell polynomials generating function (1)Thus replacing 119909 by the multiplicative operator
119901of the
2VgP 119901119899(119909 119910) in the lhs of (1) and denoting the resultant
2VgAP by119901119860119899(119909 119910) we have
119860 (119905) 119890(119901119905)
=
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (32)
Now using (31) in the exponential term in the lhs of(32) we get the generating function for
119901119860119899(119909 119910) as
119860 (119905) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (33)
In view of (5) generating function (33) can be expressedequivalently as
1
119892 (119905)119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (34)
Now we frame the 2VgAP119901119860119899(119909 119910) within the context
of monomiality principle formalism We prove the followingresults
Theorem 4 The 2VgAP119901119860119899(119909 119910) are quasimonomial with
respect to the following multiplicative and derivative operators
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
(35a)
or equivalently
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
minus1198921015840
(119863119909)
119892 (119863119909)
(35b)
119901119860
= 119863119909 (36)
respectively
Proof Differentiating (33) partially with respect to 119905 we find
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)+
1198601015840
(119905)
119860 (119905)) 119860 (119905) 119890
119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(37)
Since 119860(119905) and 120601(119910 119905) are invertible series of 119905 therefore1198601015840
(119905)119860(119905) and 1206011015840
(119910 119905)120601(119910 119905) possess power series expan-sions of 119905 Thus in view of identity (22) we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119860 (119905) 119890119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(38)
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 International Journal of Analysis
Table2Con
tinued
SNo
119892(119905
)119860
(119905)
119872119867(119898)119860119875119867(119898)119860
Generatingfunctio
nsPo
lyno
mials
(VIII)
(119905
120582119890119905
minus1
)minus120572
(119905
120582119890119905
minus1
)120572
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(120572minus
119863119909)
minus120572
119863119909
(120582119890119863119909
minus1)
119863119909
(119905
120582119890119905
minus1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898120572)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mialsof
order120572
(IX)
120582119890119905
minus1
119905
119905
120582119890119905
minus1
119909+
119898119910
119863119898minus1
119909+
120582119890119863119909
(1minus
119863119909)
minus1
119863119909
(120582119890119863119909
minus1)
119863119909
119905
120582119890119905
minus1
119890119909119905+119910119905119898
=infin sum119899=0
119867B(119898)
119899(119909
119910120582
)119905119899 119899
TheG
ould-H
opper-A
posto
l-Berno
ulli
polyno
mials
(X)
(2
120582119890119905
+1
)minus120572
(2
120582119890119905
+1
)120572
119909+
119898119910
119863119898minus1
119909minus
120582120572
119890119863119909
(120582119890119863119909
+1)119863119909
(2
120582119890119905
+1
)120572
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898120572)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
1120572
=1
TheG
ould-H
opper-A
posto
l-Euler
polyno
mialsof
order120572
(XI)
120582119890119905
+1
2
2
120582119890119905
+1
119909+
119898119910
119863119898minus1
119909minus
120582119890119863119909
(120582119890119863119909
+1)119863119909
2
120582119890119905
+1
119890119909119905+119910119905119898
=infin sum119899=0
119867E(119898)
119899(119909
119910120582
)119905119899 1198991003816 1003816 1003816 1003816
119905+log
1205821003816 1003816 1003816 1003816
lt120587
TheG
ould-H
opper-A
posto
l-Euler
polyno
mials
(XII)
119890minus(sum119903+1
ℎ=0120585ℎ119905ℎ)
119890(sum119903+1
ℎ=0120585ℎ119905ℎ)
120585 119903+1
=0
119909+
119898119910
119863119898minus1
119909+119903+1
sumℎ=1
ℎ120585 ℎ
119863ℎminus1
119909119863119909
119890(sum119903+1
ℎ=0120585ℎ119905ℎ+119909119905+119910119905119898)
=infin sum119899=0
119867119860(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-generalized
Gou
ld-H
opperp
olyn
omials(fo
r119903=
1
theG
ould-H
opper-Hermite
polyno
mials119867
119867(119898)
119899(119909
119910)andfor
119903=
2the
Gou
ld-H
opperc
lassical
2-orthogon
alpo
lyno
mials)
(XIII)
(1minus
119905 )119904+1
1
(1minus
119905 )119904+1
119909+
119898119910
119863119898minus1
119909minus
119898+
1
1minus
119863119909
119863119909
1
(1minus
119905 )119904+1119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898119904)
119899(119909
119910)
119905119899
TheG
ould-H
opper-Miller-Lee
polyno
mials(fo
r119904=
0the
Gou
ld-H
opper-trun
catedexpo
nential
polyno
mials119867
119890119898 119899(119909
119910)andfor
119904=
120573minus
1Gou
ld-H
opper-mod
ified
Laguerre
polyno
mials119867
119891(119898120573)
119899(119909
119910))
(XIV
)(119890119905
+1)
2119905
2119905
(119890119905+
1)
119909+
119898119910
119863119898minus1
119909+
119890119863119909
(1minus
119863119909)
+1
(119890119863119909
+1)
119863119909
2119905
(119890119905+
1)119890119909119905+119910119905119898
=infin sum119899=0
119867119866(119898)
119899(119909
119910)
119905119899 119899
TheG
ould-H
opper-Genocchi
polyno
mials
International Journal of Analysis 7
If 120601(119910 119905) is an invertible series and 1206011015840
(119910 119905)120601(119910 119905) hasTaylorrsquos series expansion in powers of 119905 then in view of theidentity
119863119909
119890119909119905
120601 (119910 119905) = 119905 (119890119909119905
120601 (119910 119905)) (22)
we can write
1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119890119909119905
120601 (119910 119905) =1206011015840
(119910 119905)
120601 (119910 119905)(119890119909119905
120601 (119910 119905)) (23)
Now using (23) in the lhs of (21) we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (24)
Making use of generating function (14) in the lhs of theabove equation we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899
(25)
which on equating the coefficients of like powers of 119905 in bothsides gives
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119901119899
(119909 119910) = 119901119899+1
(119909 119910) (26)
Thus in view of monomiality principle equation (6) theabove equation yields assertion (19) of Lemma 1 Again usingidentity (22) in (14) we have
119863119909
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119899minus1
(119909 119910)119905119899
(119899 minus 1) (27)
Equating the coefficients of like powers of 119905 in both sidesof (27) we find
119863119909
119901119899
(119909 119910) = 119899119901119899minus1
(119909 119910) (119899 ge 1) (28)
which in view of monomiality principle equation (7) yieldsassertion (20) of Lemma 1
Remark 2 The operators given by (19) and (20) satisfycommutation relation (8) Also using expressions (19) and(20) in (9) we get the following differential equation satisfiedby 2VgP 119901
119899(119909 119910)
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus 119899) 119901119899
(119909 119910) = 0 (29)
Remark 3 Since 1199010(119909 119910) = 1 therefore in view of monomi-
ality principle equation (10) we have
119901119899
(119909 119910) = (119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
119899
1 (1199010
(119909 119910) = 1) (30)
Also in view of (11) (14) and (19) we have
exp (119901119905) 1 = 119890
119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 (31)
Now we proceed to introduce the 2-variable general-Appell polynomials (2VgAP) In order to derive the gener-ating functions for the 2VgAP we take the 2VgP 119901
119899(119909 119910) as
the base in the Appell polynomials generating function (1)Thus replacing 119909 by the multiplicative operator
119901of the
2VgP 119901119899(119909 119910) in the lhs of (1) and denoting the resultant
2VgAP by119901119860119899(119909 119910) we have
119860 (119905) 119890(119901119905)
=
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (32)
Now using (31) in the exponential term in the lhs of(32) we get the generating function for
119901119860119899(119909 119910) as
119860 (119905) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (33)
In view of (5) generating function (33) can be expressedequivalently as
1
119892 (119905)119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (34)
Now we frame the 2VgAP119901119860119899(119909 119910) within the context
of monomiality principle formalism We prove the followingresults
Theorem 4 The 2VgAP119901119860119899(119909 119910) are quasimonomial with
respect to the following multiplicative and derivative operators
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
(35a)
or equivalently
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
minus1198921015840
(119863119909)
119892 (119863119909)
(35b)
119901119860
= 119863119909 (36)
respectively
Proof Differentiating (33) partially with respect to 119905 we find
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)+
1198601015840
(119905)
119860 (119905)) 119860 (119905) 119890
119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(37)
Since 119860(119905) and 120601(119910 119905) are invertible series of 119905 therefore1198601015840
(119905)119860(119905) and 1206011015840
(119910 119905)120601(119910 119905) possess power series expan-sions of 119905 Thus in view of identity (22) we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119860 (119905) 119890119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(38)
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 7
If 120601(119910 119905) is an invertible series and 1206011015840
(119910 119905)120601(119910 119905) hasTaylorrsquos series expansion in powers of 119905 then in view of theidentity
119863119909
119890119909119905
120601 (119910 119905) = 119905 (119890119909119905
120601 (119910 119905)) (22)
we can write
1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119890119909119905
120601 (119910 119905) =1206011015840
(119910 119905)
120601 (119910 119905)(119890119909119905
120601 (119910 119905)) (23)
Now using (23) in the lhs of (21) we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899 (24)
Making use of generating function (14) in the lhs of theabove equation we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=0
119901119899+1
(119909 119910)119905119899
119899
(25)
which on equating the coefficients of like powers of 119905 in bothsides gives
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
) 119901119899
(119909 119910) = 119901119899+1
(119909 119910) (26)
Thus in view of monomiality principle equation (6) theabove equation yields assertion (19) of Lemma 1 Again usingidentity (22) in (14) we have
119863119909
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119899minus1
(119909 119910)119905119899
(119899 minus 1) (27)
Equating the coefficients of like powers of 119905 in both sidesof (27) we find
119863119909
119901119899
(119909 119910) = 119899119901119899minus1
(119909 119910) (119899 ge 1) (28)
which in view of monomiality principle equation (7) yieldsassertion (20) of Lemma 1
Remark 2 The operators given by (19) and (20) satisfycommutation relation (8) Also using expressions (19) and(20) in (9) we get the following differential equation satisfiedby 2VgP 119901
119899(119909 119910)
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus 119899) 119901119899
(119909 119910) = 0 (29)
Remark 3 Since 1199010(119909 119910) = 1 therefore in view of monomi-
ality principle equation (10) we have
119901119899
(119909 119910) = (119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
)
119899
1 (1199010
(119909 119910) = 1) (30)
Also in view of (11) (14) and (19) we have
exp (119901119905) 1 = 119890
119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119899
(119909 119910)119905119899
119899 (31)
Now we proceed to introduce the 2-variable general-Appell polynomials (2VgAP) In order to derive the gener-ating functions for the 2VgAP we take the 2VgP 119901
119899(119909 119910) as
the base in the Appell polynomials generating function (1)Thus replacing 119909 by the multiplicative operator
119901of the
2VgP 119901119899(119909 119910) in the lhs of (1) and denoting the resultant
2VgAP by119901119860119899(119909 119910) we have
119860 (119905) 119890(119901119905)
=
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (32)
Now using (31) in the exponential term in the lhs of(32) we get the generating function for
119901119860119899(119909 119910) as
119860 (119905) 119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (33)
In view of (5) generating function (33) can be expressedequivalently as
1
119892 (119905)119890119909119905
120601 (119910 119905) =
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 (34)
Now we frame the 2VgAP119901119860119899(119909 119910) within the context
of monomiality principle formalism We prove the followingresults
Theorem 4 The 2VgAP119901119860119899(119909 119910) are quasimonomial with
respect to the following multiplicative and derivative operators
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
(35a)
or equivalently
119901119860
= 119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
minus1198921015840
(119863119909)
119892 (119863119909)
(35b)
119901119860
= 119863119909 (36)
respectively
Proof Differentiating (33) partially with respect to 119905 we find
(119909 +1206011015840
(119910 119905)
120601 (119910 119905)+
1198601015840
(119905)
119860 (119905)) 119860 (119905) 119890
119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(37)
Since 119860(119905) and 120601(119910 119905) are invertible series of 119905 therefore1198601015840
(119905)119860(119905) and 1206011015840
(119910 119905)120601(119910 119905) possess power series expan-sions of 119905 Thus in view of identity (22) we have
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119860 (119905) 119890119909119905
120601 (119910 119905)
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(38)
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Journal of
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Analysis
which on using generating function (33) becomes
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
)
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(39)
or equivalentlyinfin
sum119899=0
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910)119905119899
119899
=
infin
sum119899=0
119901119860119899+1
(119909 119910)119905119899
119899
(40)
Now equating the coefficients of like powers of 119905 in theabove equation we find
(119909 +1206011015840
(119910 119863119909)
120601 (119910 119863119909)
+1198601015840
(119863119909)
119860 (119863119909)
) 119901119860119899
(119909 119910) =119901119860119899+1
(119909 119910)
(41)
which in view of (6) yields assertion (35a) of Theorem 4Also in view of relation (5) assertion (35a) can be expressedequivalently as (35b)
Again in view of identity (22) we have
119863119909
119860 (119905) 119890119909119905
120601 (119910 119905) = 119905119860 (119905) 119890119909119905
120601 (119910 119905) (42)
which on using generating function (33) becomes
119863119909
infin
sum119899=0
119901119860119899
(119909 119910)119905119899
119899 =
infin
sum119899=1
119901119860119899minus1
(119909 119910)119905119899
(119899 minus 1) (43)
Equating the coefficients of like powers of 119905 in the aboveequation we find
119863119909
119901119860119899
(119909 119910) = 119899119901119860119899minus1
(119909 119910) (119899 ge 1) (44)
which in view of (7) yields assertion (36) ofTheorem 4
Theorem 5 The 2VgAP119901119860119899
(119909 119910) satisfy the following differ-ential equations
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
+1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45a)
or equivalently
(119909119863119909
+1206011015840
(119910 119863119909)
120601 (119910 119863119909)
119863119909
minus1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119901119860119899
(119909 119910) = 0
(45b)
Proof Using (35a) and (36) in (9) we get assertion (45a) Fur-ther using (35b) and (36) in (9) we get assertion (45b)
Note 1 With the help of the results derived above and bytaking 119860(119905) (or 119892(119905)) of the Appell polynomials listed inTable 1 we can derive the generating function and otherresults for the members belonging to 2VgAP family
3 Examples
We consider examples of certain members belonging to the2VgAP family
Taking 120601(119910 119905) = 119890119910119905119898
(that is when the 2VgP 119901119899(119909 119910)
reduces to the GHP 119867(119898)
119899(119909 119910)) in generating function
(33) we find that the Gould-Hopper-Appell polynomials(GHAP)
119867119860(119898)
119899(119909 119910) are defined by the following generating
function
119860 (119905) 119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899(46)
or equivalently
1
119892 (119905)119890(119909119905+119910119905
119898)
=
infin
sum119899=0
119867119860(119898)
119899(119909 119910)
119905119899
119899 (47)
Using (1) in (46) (or (3) in (47)) we get the followingseries definition for
119867119860(119898)
119899(119909 119910) in terms of the Appell
polynomials 119860119899(119909)
119867119860(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119860119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (48)
In view of (35a) (35b) and (36) we note that the GHAP119867
119860(119898)
119899(119909 119910) are quasimonomial under the action of the
following multiplicative and derivative operators
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909+
1198601015840
(119863119909)
119860 (119863119909)
(49a)
or equivalently
119867(119898)119860
= 119909 + 119898119910119863119898minus1
119909minus
1198921015840
(119863119909)
119892 (119863119909)
(49b)
119867(119898)119860
= 119863119909 (50)
respectively Also in view of (45a) and (45b) we find thatthe GHAP
119867119860(119898)
119899(119909 119910) satisfy the following differential
equation
(119909119863119909
+ 119898119910119863119898
119909+
1198601015840
(119863119909)
119860 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51a)
or equivalently
(119909119863119909
+ 119898119910119863119898
119909minus
1198921015840
(119863119909)
119892 (119863119909)
119863119909
minus 119899)119867
119860(119898)
119899(119909 119910) = 0
(51b)
Remark 6 In view of (16) and (17) we note that for 119898 =
2 the GHAP119867
119860(119898)
119899(119909 119910) reduce to the Hermite-Appell
polynomials (HAP)119867
119860119899(119909 119910) Therefore taking 119898 = 2 in
(46) (47) (48) (49a) (49b) (50) (51a) and (51b) we get thecorresponding results for the HAP
119867119860119899(119909 119910)
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 9
Remark 7 In view of (16) we note that the 2VHKdFP119867119899(119909 119910) are related to the classical Hermite polynomials
119867119899(119909) [11] or 119867119890
119899(119909) as
119867119899
(2119909 minus1) = 119867119899
(119909)
119867119899
(119909 minus1
2) = 119867119890
119899(119909)
(52)
Therefore taking 119910 = minus1 and replacing 119909 by 2119909 (or taking119910 = minus12) in (46)ndash(51b) we get the corresponding resultsfor the classical Hermite-Appell polynomials
119867119860119899(119909) (or
119867119890119860119899(119909))
There are several members belonging to 2VgP familyThus the results for the corresponding 2VgAP can be obtainedby taking other examples We present the list of somemembers belonging to the GHAP family in Table 2
Note 2 Since for 119898 = 2 the GHAP119867
119860119898
119899(119909 119910) reduce to the
HAP119867
119860119899(119909 119910) therefore for 119898 = 2 Table 2 gives the list of
the corresponding HAP119867
119860119899(119909 119910)
The results established in this paper are general andinclude new families of special polynomials consequentlyintroducing many new special polynomials
Appendix
New classes of Bernoulli numbers and polynomials areintroduced which are used to evaluate partial sums involv-ing other special polynomials see for example [21 22]Here we consider the Gould-Hopper-Bernoulli polyno-mials (GHBP) Gould-Hopper-Euler polynomials (GHEP)Hermite-Bernoulli polynomials (HBP) Hermite-Euler poly-nomials (HEP) classical Hermite-Bernoulli polynomials(cHBP) and classical Hermite-Euler polynomials (cHEP)We give the surface plots of these polynomials for suitablevalues of the parameters and indices Also we give the graphsof the corresponding single-variable polynomials
The GHBP119867
119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) and HEP
119867119864119899(119909 119910) are defined by the following
series
119867119861(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119861119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A1)
119867119864(119898)
119899(119909 119910) = 119899
[119899119898]
sum119903=0
119864119899minus119898119903
(119909) 119910119903
(119899 minus 119898119903)119903 (A2)
119867119861119899
(119909 119910) = 119899
[1198992]
sum119903=0
119861119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A3)
119867119864119899
(119909 119910) = 119899
[1198992]
sum119903=0
119864119899minus2119903
(119909) 119910119903
(119899 minus 2119903)119903 (A4)
respectively Taking 119910 = minus12 in (A3) and (A4) we get theseries definitions for the cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909)
as
119867119861119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119861119899minus2119903
(119909)
(119899 minus 2119903)119903 (A5)
119867119864119899
(119909) = 119899
[1198992]
sum119903=0
(minus1
2)119903 119864119899minus2119903
(119909)
(119899 minus 2119903)119903 (A6)
respectivelyTo draw the surface plots of these polynomials we use
the values of the Bernoulli polynomials 119861119899(119909) and the Euler
polynomials 119864119899(119909) for 119899 = 0 1 2 3 4 and 5 We give the list
of the first fewBernoulli and the Euler polynomials in Table 3Now we consider the GHBP
119867119861119898
119899(119909 119910) GHEP
119867119864119898
119899(119909 119910) HBP
119867119861119899(119909 119910) HEP
119867119864119899(119909 119910) cHBP
119867119890119861119899(119909) and cHEP
119867119890119864119899(119909) for 119898 = 3 and 119899 = 5 so that
we have
119867119861(3)
5(119909 119910) = 119861
5(119909) + 60119861
2(119909) 119910 (A7)
119867119864(3)
5(119909 119910) = 119864
5(119909) + 60119864
2(119909) 119910 (A8)
1198671198615
(119909 119910) = 1198615
(119909) + 201198613
(119909) 119910 + 601198611
(119909) 1199102
(A9)
1198671198645
(119909 119910) = 1198645
(119909) + 201198643
(119909) 119910 + 601198641
(119909) 1199102
(A10)
1198671198901198615
(119909) = 1198615
(119909) minus 101198613
(119909) + 151198611
(119909) (A11)
1198671198901198645
(119909) = 1198645
(119909) minus 101198643
(119909) + 151198641
(119909) (A12)
respectively Using the particular values of 119861119899(119909) and 119864
119899(119909)
given in Table 3 we find
119867119861(3)
5(119909 119910)=119909
5
minus5
21199094
+5
31199093
+601199092
119910minus60119909119910minus1
6119909+
1
10119910
(A13)
119867119864(3)
5(119909 119910)=119909
5
minus5
21199094
+601199092
119910+5
31199092
minus60119909119910minus1
2+
1
10119910
(A14)
119867
1198615
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 +5
31199093
+ 601199091199102
minus 601199092
119910 + 10119909119910 minus 301199102
minus1
6119909
(A15)
1198671198645
(119909 119910) = 1199095
minus5
21199094
+ 201199093
119910 + 601199091199102
minus 301199092
119910 +5
31199092
minus 301199102
+10
3119910 minus
1
2
(A16)
1198671198901198615
(119909) = 1199095
minus5
21199094
minus25
31199093
+ 151199092
+59
6119909 minus
15
2 (A17)
1198671198901198645
(119909) = 1199095
minus5
21199094
minus 101199093
+50
31199092
+ 15119909 minus29
3 (A18)
respectively
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 International Journal of Analysis
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(a)
0
5
0
5
0
05
1
minus1
minus05
minus05
minus05
minus15
times104
(b)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(c)
0
5
0
5
0
1
2
minus3
minus2
minus1
minus5 minus5
times104
(d)
minus5 0 5minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(e)minus5 0 5
minus3500
minus3000
minus2500
minus2000
minus1500
minus1000
minus500
0
500
1000
(f)
Figure 1
Table 3 List of the first few Bernoulli and the Euler polynomials
119899 0 1 2 3 4 5
119861119899(119909) 1 119909 minus
1
21199092
minus 119909 +1
61199093
minus3
21199092
+119909
21199094
minus 21199093
+ 1199092
minus1
301199095
minus5
21199094
+5
31199093
minus119909
6
119864119899(119909) 1 119909 minus
1
21199092
minus 119909 1199093
minus3
21199092
+1
61199094
minus 21199093
+2
3119909 119909
5
minus5
21199094
+5
31199092
minus1
2
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Analysis 11
In view of equations (A13)ndash(A18) we get Figure 1
Acknowledgments
The authors are thankful to the anonymous referee for usefulsuggestions towards the improvement of the paperThis workhas been done under the Senior Research Fellowship (OfficeMemo no AcadD-1562MR) sanctioned to the secondauthor by the University Grants Commission Governmentof India New Delhi
References
[1] P Appell ldquoSur une classe de polynomesrdquo Annales Scientifiquesde lrsquoEcole Normale Superieure vol 9 pp 119ndash144 1880
[2] S Roman The Umbral Calculus vol 111 of Pure and AppliedMathematics Academic Press New York NY USA 1984
[3] E D Rainville Special Functions Macmillan New York NYUSA 1960 reprinted by Chelsea Bronx NY USA 1971
[4] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol III McGraw-Hill NewYork NY USA 1955
[5] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions Vol II McGraw-Hill NewYork NY USA 1953
[6] G Bretti P Natalini and P E Ricci ldquoGeneralizations ofthe Bernoulli and Appell polynomialsrdquo Abstract and AppliedAnalysis vol 2004 no 7 pp 613ndash623 2004
[7] Q-M Luo and H M Srivastava ldquoSome generalizations of theApostol-Bernoulli and Apostol-Euler polynomialsrdquo Journal ofMathematical Analysis and Applications vol 308 no 1 pp 290ndash302 2005
[8] T M Apostol ldquoOn the Lerch zeta functionrdquo Pacific Journal ofMathematics vol 1 pp 161ndash167 1951
[9] Q-M Luo ldquoApostol-Euler polynomials of higher order andGaussian hypergeometric functionsrdquo Taiwanese Journal ofMathematics vol 10 no 4 pp 917ndash925 2006
[10] K Douak ldquoThe relation of the 119889-orthogonal polynomials tothe Appell polynomialsrdquo Journal of Computational and AppliedMathematics vol 70 no 2 pp 279ndash295 1996
[11] L C Andrews Special Functions for Engineers and AppliedMathematicians Macmillan New York NY USA 1985
[12] G Dattoli S Lorenzutta and D Sacchetti ldquoIntegral represen-tations of new families of polynomialsrdquo Italian Journal of Pureand Applied Mathematics no 15 pp 19ndash28 2004
[13] W Magnus F Oberhettinger and R P Soni Formulas andTheorems for the Special Functions of Mathematical Physicsvol 52 of Die Grundlehren der mathematischen WissenschaftenSpringer New York NY USA 3rd edition 1966
[14] G Dattoli M Migliorati and H M Srivastava ldquoSheffer poly-nomials monomiality principle algebraic methods and thetheory of classical polynomialsrdquo Mathematical and ComputerModelling vol 45 no 9-10 pp 1033ndash1041 2007
[15] J F Steffensen ldquoThe poweroid an extension of the mathemat-ical notion of powerrdquo Acta Mathematica vol 73 pp 333ndash3661941
[16] G Dattoli ldquoHermite-Bessel and Laguerre-Bessel functions aby-product of the monomiality principlerdquo in Advanced SpecialFunctions and Applications (Melfi 1999) vol 1 of ProcMelfi SchAdv Top Math Phys pp 147ndash164 Aracne Rome Italy 2000
[17] G Bretti C Cesarano and P E Ricci ldquoLaguerre-type expo-nentials and generalized Appell polynomialsrdquo Computers ampMathematics with Applications vol 48 no 5-6 pp 833ndash8392004
[18] P Appell and J Kampe de Feriet Fonctions Hypergeometriqueset Hyperspheriques Polynomes drsquo Hermite Gauthier-VillarsParis France 1926
[19] H W Gould and A T Hopper ldquoOperational formulas con-nected with two generalizations of Hermite polynomialsrdquoDukeMathematical Journal vol 29 pp 51ndash63 1962
[20] S Khan G Yasmin R Khan and N A M Hassan ldquoHermite-based Appell polynomials properties and applicationsrdquo Journalof Mathematical Analysis and Applications vol 351 no 2 pp756ndash764 2009
[21] G Dattoli C Cesarano and S Lorenzutta ldquoBernoulli numbersand polynomials from amore general point of viewrdquo RendicontidiMatematica e delle sueApplicazioni vol 22 pp 193ndash202 2002
[22] G Dattoli S Lorenzutta and C Cesarano ldquoFinite sums andgeneralized forms of Bernoulli polynomialsrdquo Rendiconti diMatematica e delle sue Applicazioni vol 19 no 3 pp 385ndash3911999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
![Kent Academic RepositoryPhD]AnaLoureirothesis.pdf · Hahn’s generalised problem and corresponding Appell polynomial sequences ... introduzimos e desenvolvemos o conceito ... nous](https://static.fdocuments.us/doc/165x107/5c0c3e8209d3f2e4358bd642/kent-academic-repository-phd-hahns-generalised-problem-and-corresponding.jpg)
![THE ROLE OF THE APPELL TRANSFORMATION IN THE THEORY … · Appell transform is (1.1) A[v] =^,(M = /c(x, t)v(j, =1),-*»/4« (1.2) k(x, t) = (4ni)1/2 Here k(x, t) is the fundamental](https://static.fdocuments.us/doc/165x107/5f5391351fb1576b711386d5/the-role-of-the-appell-transformation-in-the-theory-appell-transform-is-11-av.jpg)