Applied Mathematics, 2014, 5, 1573-1585 Published Online June 2014 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2014.510150

How to cite this paper: El-Desouky, B.S., El-Bedwehy, N.A., Mustafa, A. and Menem, F.M.A. (2014) A Family of Generalized Stirling Numbers of the First Kind. Applied Mathematics, 5, 1573-1585. http://dx.doi.org/10.4236/am.2014.510150

A Family of Generalized Stirling Numbers of the First Kind Beih S. El-Desouky1, Nabela A. El-Bedwehy2, Abdelfattah Mustafa1, Fatma M. Abdel Menem2 1Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt 2Mathematics Department, Faculty of Science, Damietta University, Damietta, Egypt Email: b_desouky@yahoo.com Received 4 March 2014; revised 4 April 2014; accepted 11 April 2014

Copyright © 2014 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract A modified approach via differential operator is given to derive a new family of generalized Stirl-ing numbers of the first kind. This approach gives us an extension of the techniques given by El-Desouky [1] and Gould [2]. Some new combinatorial identities and many relations between dif-ferent types of Stirling numbers are found. Furthermore, some interesting special cases of the ge-neralized Stirling numbers of the first kind are deduced. Also, a connection between these num-bers and the generalized harmonic numbers is derived. Finally, some applications in coherent states and matrix representation of some results obtained are given.

Keywords Stirling Numbers, Comtet Numbers, Creation, Annihilation, Differential Operator, Maple Program

1. Introduction Gould [2] proved that

( ) ( ) ( ) ( )11 1

de e 1 , e , , ,d

n nn n kx nx k nx kx

k kD s n k D s n k D D D

x−

= =

= − = = =∑ ∑ (1)

where ( ),s n k and ( )1 ,s n k are the usual Stirling numbers and the singles Stirling numbers of the first kind, respectively, defined by

( ) ( ) ( ) ( ),00

, , ,0 and , 0 for .n

knn

kx s n k x s n s n k k nδ

=

= = = >∑ (2)

( ) ( ) ( )1 1 ,0 10

, , ,0 and , 0 for ,n

knn

kx s n k x s n s n k k nδ

=

= = = >∑ (3)

B. S. El-Desouky et al.

1574

( ) ( ) ( )where 1 1nx x x x n= − − + and ( ) ( )1 1x x x x n= + + − . These numbers satisfy the recurrence relations

( ) ( ) ( )1, , 1 , ,s n k s n k ns n k+ = − − (4)

( ) ( ) ( )1 1 11, , 1 , .s n k s n k ns n k+ = − + (5)

EL-Desouky [1] defined the generalized Stirling numbers of the first kind ( ); , ,s k r s called ( ),r s -Stirling numbers of the first kind by

( ) ( )1 2

12 2 1 1

1

e e e e ; , ,n n

lln ns s sr xr x s r x s r x s k

k sD D D s k D=

+ + +

=

∑= ∑

r s (6)

( ); , 0s k =r s for 1k s< or 1

njjk s

=> ∑ and ( )0; , 1,s =r s where ( )1 2: , , , nr r r= r is a sequence of real

numbers and ( )1 2: , , , ns s s= s is a sequence of nonnegative integers. Equation (6) is equivalent to

( ) ( )1 2

12 2 1 1

1

e e e e ; , ,n n

lln ns s sr ar a s r a s r a s k

k sa a a s k a

++ + + =+ + +

=

∑= ∑

r s (7)

where a+ and a are boson creation and annihilation operators, respectively, and satisfy the commutation rela- tion , 1.a a+ =

The numbers ( ); ,s k r s satisfy the recurrence relation

( ) ( )11

11 1

0 1; , ; , ,

nns is n

nn n j

i j

ss k r s r s k i

i

++

−+

+ += =

⊕ ⊕ = −

∑ ∑r s r s (8)

with the notations ( )1 1 2 1, , ,n nr r r r+ +⊕ = r and ( )1 1 2 1, , ,n ns s s s+ +⊕ = s . The numbers ( ); ,s k r s have the explicit formula

( )1

11

, 0 11; , ,

l

n n

in ll

jk j jl l

ss k r

iσ β−

−+

= − ≥ ==

=

∑ ∑∏r s (9)

where 0 ,nn jj iσ

== ∑ with 0 0i = and 1 .n

n jj sβ=

= ∑

Moreover El-Desouky [1] derived many special cases and some applications. For the proofs and more details, see [1].

The generalized falling factorial of x associated with the sequence ( )0 1 1, , , nα α α α −= : of order n, where

0 1 1, , , nα α α − are real numbers, is defined by ( ) ( ) ( )100, , , 1.n

in ix x xα α α−

== − =∏

Comtet [3] [4] and [5] defined ( ),s n kα the generalized Stirling numbers of the first kind, which are called Comtet numbers, by

( ) ( )0

, , .n

kn

kx s n k xαα

=

= ∑ (10)

These numbers satisfy the recurrence relation

( ) ( ) ( )1, 1, 1 1, .ns n k s n k s n kα α αα −= − − − − (11)

El-Desouky and Cakic [6] defined ( ), ;s n k sα , the generalized Comtet numbers by

( ) ( ) ( )| |1 1

00 0, ; ,

j jj jsn ns s k

jkj j

x x s n kα ααδ δ α δ

− −−

== =

= − = ∑∏ ∏ s (12)

where ( )10 , , ; 0n

jj s s n k−

== =∑ αs s for ( ) ( ) ( )1

0,s ,0; 1 jn ssjjk n α−

=> = − ∏αs s and ( )0 1, , , ns s s= s .

For more details on generalized Stirling numbers via differential operators, see [7]-[10] and [11].

B. S. El-Desouky et al.

1575

The paper is organized as follows: In Section 2, using the differential operator ( ) ( )( )2 2 2 1 1 1e e e e e en n nr x s r x r x s r x r x s r xD D D we define a new family

of generalized Stirling numbers of the first kind, denoted by ( ); ,ks r s . A recurrence relation and an explicit formula of these numbers are derived. In Section 3, some interesting special cases are discussed. Moreover some new combinatorial identities and a connection between ( ), ; ,n k r ss and the generalized harmonic numbers

( )inO are given. In Section 4, some applications in coherent states and matrix representation of some results ob-

tained are given. Section 5 is devoted to the conclusion, which handles the main results derived throughout this work. Finally, a computer program is written using Maple and executed for calculating the generalized Stirling numbers of the first kind and some special cases, see Appendix.

2. Main Results Let ( )1 2, , , nr r r=r : be a sequence of real numbers and ( )1 2, , , ns s s=s : be a sequence of nonnegative integers.

Definition 2.1 The generalized Stirlng numbers ( ); ,k r ss are defined by

( ) ( )( ) ( ) ( )12 2 2 1 1 12

0e e e e e e e ; , ,

n nlln n n

rr x s r x r x s r x r x s r x k

kD D D k D

β=

=

∑= ∑ r s s (13)

where 1 ,nn jj sβ

== ∑ ( ); , 0k =r ss for nk β> and ( )0; , 1=r ss .

Equation (13) is equivalent to

( ) ( )( ) ( ) ( )12 2 2 1 1 12

0e e e e e e e ; , .

nlln n n

r ar a s r a r a s r a r a s r a k

ka a a k a

β++ + + + + + =

=

∑= ∑ r s s (14)

Theorem 2.1 The numbers ( ); ,k r ss satisfy the recurrence relation

( ) ( )11

11 1 1

0 1; , 2 ; , ,

nns is n

nn n j n

i j

sk r s r r k i

i

++

−+

+ + += =

⊕ ⊕ = + −

∑ ∑r s r ss s (15)

with the notations ( ) ( )1 1 2 1 1 1 2 1, , , and , , , .n n n nr r r r s s s s+ + + +⊕ = ⊕ =r s : : Proof

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( )

1 11 11 1 1

111 1

1

1

1

2 2

0 0

2

0

21

0 1

2 1

e ; , e e e ; ,

e e ; ,

e 2 ; ,

e

n nn nj jj jn n n

n nj njn n

nn n

jj

njj

r x r xr x s r xk k

k k

r r xr x s k

k

snr x m

j nm j

r x n

k D D k D

D k D

D r r m D

si

β β

β

β

+ += =+ + +

+=+ +

+

=

=

= =

+

=

+= =

+

∑ ∑

∑

∑

∑

=

=

= + +

=

∑ ∑

∑

∑ ∑

r s r s

r s

r s

s s

s

s

( )

( ) ( )

11

11 11

10 0 1

2 11

0 0 1

2 ; ,

e 2 ; , .

nn n

nn n njj

s is nm i

j nm i j

s is nr x n kj n

k i j

r r m D

sr r k i D

i

β

β

++

++ +

=

−

++

= = =

−+

+= = =

∑

+

= + −

∑∑ ∑

∑∑ ∑

r s

r s

s

s

Equating the coefficients of kD on both sides yields (15). Theorem 2.2 The numbers ( ); ,k r ss have the explicit formula

( )1

0, 0 0 11

; , 2 , where , 0.l

jn n

in l n

ij l n

k j j jl

sk r r i r

iσ βσ

−

= − ≥ = ==

= + = =

∑ ∑ ∑∏r ss (16)

B. S. El-Desouky et al.

1576

Proof

( ) ( )1

1 11 1 1 1 1 1 1 1

1

121 1

0 1

e e e e e ,s

s ir x s r x r x r x r x s i

i

sD D r I r D

i−

=

= + =

∑

( )( ) ( ) ( ) ( )

( ) ( ) ( )( )

( )

1 11 11 22 2 2 1 1 1 2 2 2 1 1 1 2 2 1 1

1 1

1211 22 1 1

1

1 2

2

2 1121 1

0 01 1

2 11 1 2

0 1

2 1 2

0 1 2

e e e e e e e e e

e e 2

e

s si ir r xr x s r x r x s r x r x s r x r x s i r x s s i

i i

s sir r xr x s i

i

r r x

i

ssD D D r D D r D

i i

sr D r r I D

i

s si i

+− −

= =

+ −

=

+

=

= =

= + +

=

∑ ∑

∑

( ) ( )1 2

1 2 1 2 1 2

1

1 1 20

2 ,s s

i i s s i i

ir r r D + − −

=

+∑ ∑

thus, by iteration, we get

( )

2 2 2 1 1 1

1 21 1 1

1 2

120

0 0 0 11

(e e ) (e e )(e e )

e 2 , 0.

n n n

jn n n nll j jj j

n

r x s r x r x s r x r x s r x

iss s jnr x s ijl j

i i i lj j

D D D

sr r D r

i= = =

−−

= = = ==

∑ ∑ ∑ = + =

∑ ∑ ∑ ∑∏

(17)

Setting 1 1 ,n nj j n nj js i kβ σ

= =− = − =∑ ∑ we obtain

( ) ( )( ) ( )12 2 2 1 1 112

0 , 0 11e e e e e e e 2 .

jn nlln n n

n n

ijnr x jr x s r x r x s r x r x s r x kl j

k k j lj j

sD D D r r D

i

β

σ β

=−

= = − ≥ ==

∑ = +

∑ ∑ ∑∏ (18)

Comparing (13) and (18) yields (16).

3. Special Cases Setting and , 1, ,i ir r s s i n= = = in (13), we have the following definition.

Definition 3.1 For any real number r and nonnegative integer s, let the numbers ( ), ; , ,n k r ss be defined by

( ) ( ) ( )2

0e e e , ; , ,

nsn nr xrx s rx k

kD n k r s D

=

= ∑s (19)

where ( )0,0; , 1r s =s and ( ), ; , 0n k r s =s for k ns> . Equation (19) is equivalent to

( ) ( ) ( )2

0e e e , ; , .

nsnnr ara s ra k

ka n k r s a

++ +

=

= ∑s (20)

Corollary 3.1 The numbers ( ), ; ,n k r ss satisfy the recurrence relation

( ) ( )( ) ( )0

1, ; , 2 1 , ; , .s s i

i

sn k r s n r n k i r s

i−

=

+ = + −

∑s s (21)

Proof The proof follows directly from Equation (15) by setting ir r= and , 1, 2, , .is s i n= = Corollary 3.2 The numbers ( ), ; ,n k r ss have the explicit formula

( ) ( ), 0 1

, ; , 2 1 .j

n j

nins k

ns k i j j

sn k r s r j

iσ

−

= − ≥ =

= −

∑ ∏s (22)

Proof By substituting ir r= and , 1, 2, ,is s i n= = in Equation (17), yields

B. S. El-Desouky et al.

1577

( ) ( ) ( )1

2

0 , , 1e e e 2 1 ,j n n

n

nn inr x nsrx s rx

i i s j j

sD j r D

iσ σ−

≤ ≤ =

= −

∑ ∏

then setting nns kσ− = we have

( ) ( ) ( )2

0 , 0 1e e e 2 1 ,j

n j

nnsn inr xrx s rx ns k k

k ns k i j j

sD r j D

iσ

−

= = − ≥ =

= −

∑ ∑ ∏ (23)

hence comparing Equations (19) and (23) we obtain Equation (22). Furthermore we handle the following special cases. i) If 1r = , then we have Definition 3.2

( ) ( ) ( )2

0e e e , ;1, ,

nsn n xx s x k

kD n k s D

=

= ∑s (24)

where ( )0,0;1, 1s =s and ( ), ;1, 0n k s =s for .k ns> Corollary 3.3 The numbers ( ), ;1,n k ss satisfy the recurrence relation

( ) ( ) ( )0

1, ;1, 2 1 , ;1, .s s i

i

sn k s n n k i s

i−

=

+ = + −

∑s s (25)

Proof: The proof follows directly from Equation (21) by setting 1r = . Corollary 3.4 The numbers ( ), ;1,n k ss have the explicit formula

( ) ( ), 0 1

, ;1, 2 1 .j

n j

ni

ns k i j j

sn k s j

iσ = − ≥ =

= −

∑ ∏s (26)

Proof The proof follows directly from Equation (22) by setting 1r = . ii) If 1s = , then we have Definition 3.3 The numbers ( ), ; ,1n k rs are defined by

( ) ( ) ( )2

0e e e , ; ,1 ,

nn nr xrx rx k

kD n k r D

=

= ∑s (27)

where ( ) ,0,0; ,1 nn r δ=s and ( ), ; ,1 0n k r =s for .k n> Corollary 3.5 The numbers ( ), ; ,1n k rs satisfy the triangular recurrence relation

( ) ( ) ( ) ( ), ; ,1 1, 1; ,1 2 1 1, ; ,1 .n k r n k r n r n k r= − − + − −s s s (28)

Proof The proof follows easily from (22) by setting 1s = . Corollary 3.6 The numbers ( ), ; ,1n k rs have the following explicit formula

( ) ( ){ }, 0,1 1

, ; ,1 2 1 .j

n j

nin k

n k i jn k r r j

σ

−

= − ∈ =

= −∑ ∏s (29)

Proof The proof follows from (22) by setting 1s = . Also, using the recurrence relation (28) we can find the following explicit formula.

B. S. El-Desouky et al.

1578

Theorem 3.1 The numbers ( ), ; ,1n k rs have the following explicit expression

( ) ( ){ }1

1 2

, 0,1 21

3 2 1, ; ,1 .

11 1n j

n

i i k i n

i r i r i n rn k r

ii i+ + = ∈

+ + + − = −− −

∑

s (30)

Proof

For 0k = , ( ) ( )( )( ) ( )( ) ( )( ),0; ,1 3 5 2 1 1.2.3 2 1 .nn r r r r n r r n= − = − s

For { }0,1ni ∈ , we get

( )( ) ( )

{ }

( ){ }

( ) ( ) ( )

1 1

1 1

1 2 1

0, 0,1 21 1

1 2 1

1, 0,1 21 1

, ; ,1

3 2 32 1

11 1

3 2 31

11 1

1, 1; ,1 2 1 1, ; ,1 .

n j

n j

n

i i k i n

n

i i k i n

n k r

i r i r i n rn r

ii i

i r i r i n rii i

n k r n r n k r

−

−

−

+ + = − ∈ −

−

+ + = − ∈ −

+ + + − = ⋅ − −− −

+ + + − + ⋅ −− −

= − − + − −

∑

∑

s

s s

That is the same recurrence relation (28) for the numbers ( ), ; ,1 .n k rs This completes the proof. iii) If 1r = and 1s = , then we have Definition 3.4 The numbers ( ) ( ), : , ;1,1n k n k=s s are defined by

( ) ( )2

0e e e , ,

nnx x nx k

kD n k D

=

= ∑s (31)

where ( ) ,0,0 nn δ=s and ( ), 0n k =s for .k n> Equation (31) is equivalent to

( ) ( )2

0e e e , .

nna a na k

kD n k a

+ + +

=

= ∑s (32)

Corollary 3.7 The numbers ( ),n ks satisfy the triangular recurrence relation

( ) ( ) ( ) ( )1, , 1 2 1 , .n k n k n n k+ = − + +s s s (33)

Proof The proof follows by setting 1r = in Equation (28). Corollary 3.8 The numbers ( ),n ks have the explicit formula

( ) ( ){ }, 0,1 1

, 2 1 .j

n j

ni

n k i jn k j

σ = − ∈ =

= −∑ ∏s (34)

Proof The proof follows by setting 1r = in Equation (29). Moreover ( ),n ks have the following explicit formula. Corollary 3.9 The numbers ( ),n ks have the following explicit expression

( ) ( ){ }1

1 22

, 0,1 321

1 53 2 1, .

111 1n j

n

i i k i n

i ii i nn k

iii i+ + = ∈

+ ++ + − = −−− −

∑

s (35)

Proof The proof follows by setting 1r = in (30).

B. S. El-Desouky et al.

1579

From Equations (29) and (30) (also from Equations (34) and (35)) we have the combinatorial identities

( ){ }

( ){ }1

1 2

, 0,1 , 0,1 121

3 2 12 1 .

11 1j

n j n j

nin n k

i i k i n k i jn

i r i r i n rr j

ii i σ

−

+ + = ∈ = − ∈ =

+ + + − = − −− −

∑ ∑ ∏

(36)

( ){ }

( ){ }1

1 2

, 0,1 , 0,1 121

1 3 2 12 1 .

11 1j

n j n j

nin

i i k i n k i jn

i i i nj

ii i σ+ + = ∈ = − ∈ =

+ + + − = − −− −

∑ ∑ ∏

(37)

From Equations (29) and (34) we obtain that

( ) ( ), ; ,1 , .n kn k r r n k−=s s (38)

Remark 3.1 Operating with both sides of Equation (13) on the exponential function elx , we get

( ) ( ) ( ) ( )1 21 1 2 1 12 2 2 ; , .

nnss s k

n nk s

l r l r r l r r r k lβ

−=

+ + + + + + + + + = ∑ r s s

Therefore, since a nonzero polynomial can have only a finite set of zeros, we have

( )11

1 00 00

2 ; , , 0.j n

sjnk

m jm kj

r r x k x rβ+−

+= ==

+ + = =

∑ ∑∏ r ss (39)

If 1x = , we obtain

( )11

10 00

; , 1 2 .jn

sjn

m jk mj

k r rβ +−

+= ==

= + +

∑ ∑∏r ss (40)

Remark 3.2 From relation (39), by replacing js with 1js − , and relation (18) we conclude that

( ) ( ) 10

; , , ; , where 2 , 0,1, , 1.i

i m im

k s n k r r i nα +=

= = − + = − ∑r s s

s α (41)

This gives us a connection between ( ); ,k r ss and ( ), ; ,s n k sα the generalized Comtet numbers, see [6]. Setting ir r= and , 1, , ,is s i n= = in (39), we get

( )( ) ( )1

002 1 , ; , ,

n nss k

kjj r x n k r s x

−

==

+ + = ∑∏ s (42)

hence, we have ( ) ( ), ; , , ; ,n k r s s n k s=s α where ( )2 1 , 0,1, , 1,i i r i nα = − + = − see [6]. If 1x = , then

( ) ( )( )1

0 0, ; , 2 1 1 , 1.

nns s

k jn k r s j r n

−

= =

= + + ≥∑ ∏s (43)

Next we discuss the following special cases of (42) and (43): i) If 1r = , then

( )( ) ( )

( ) ( )( )

1

00

1

0 0

2 1 , ;1, ,

, ;1, 2 1 1 , 1,

n nss k

kj

nns s

k j

j x n k s x

n k s j n

−

==

−

= =

+ + = = + + ≥

∑∏

∑ ∏

s

s

(44)

hence we have ( ) ( ), ;1, , ,n k s s n k=s α the generalized Comtet numbers, where ( )2 1 ,i iα = − + 0,1, , 1,i n= − see [6].

ii) If 1s = , then we have

B. S. El-Desouky et al.

1580

( )( ) ( )

( ) ( )( )

1

00

1

0 0

2 1 , ; ,1 ,

, ; ,1 2 1 1 , 1.

n nk

kj

nn

k j

j r x n k r x

n k r j r n

−

==

−

= =

+ + = = + + ≥

∑∏

∑ ∏

s

s (45)

hence we obtain ( ) ( ), ; , , ,n k r s s n kα=s Comtet numbers, where ( )2 1 , 0,1, , 1i i r i nα = − + = − , see [3] and [4].

For example if 3, 2n r= = and s = 2 in (43) we have

( ) ( )26 2

0 03, ;2, 2 4 3 .

k jk j

= =

= +∑ ∏s (46)

Using Table 2, L.H.S. of (46) = s(3,0;2,2) + s(3,1;2,2) + s(3,2;2,2) + s(3,3;2,2) + s(3,4;2,2) + s(3,5;2,2) + s(3,6;2,2) = 14400

+ 22080 +12784 + 3552 + 508 + 36 + 1 = 53361. R.H.S. of (46) = ( ) ( )( )( )0

2 2 22 24 3 3 7 11 5 3361j j=

+ = =∏ .

This confirms (46) and hence (43). Another example if n = 2, r = 2 and s = 3 in (43) we have

( ) ( )16 3

0 02, ;2,3 4 3 .

k jk j

= =

= +∑ ∏s (47)

Using Table 3, L.H.S. of (47) = s(2,0;2,3) + s(2,1;2,3) + s(2,2;2,3) + s(2,3;2,3) + s(2,4;2,3) + s(2,5;2,3) + s(2,6;2,3) = 1728 +

3456 + 2736 + 1088 + 228 + 24 + 1 = 9261. R.H.S. of (46) = ( ) ( )( )1 3 3 3

0 4 3 3 7 9261j j=

+ = =∏ .

This confirms (43). iii) If 1r s= = , then we get

( )( ) ( )

( ) ( )

1

00

1

0 0

2 1 , ,

, 2 1 2 !, 1,

n nk

kj

nnn

k j

j x n k x

n k j n n

−

==

−

= =

+ + = = + = ≥

∑∏

∑ ∏

s

s

(48)

hence we have ( ) ( ), , ,n k s n k=s α which is a special case of Comtet numbers, where ( )2 1 , 0,1, , 1,i i i nα = − + = − see [3] and [4] and Table 1.

Setting ex t= , we have ( ): d d d d ,t tD x t t tD δ= = = = then substituting in (2.1) it becomes

( ) ( )( ) ( ) ( )12 2 2 1 1 12

0e ; , .

n nlln n n

r xr s r r s r r s r kt t t t

kt t t t t t k

β

δ δ δ δ=

=

∑= ∑ r s s (49)

Using, see [12],

( ) ( )( ) ( ) ( ) ,F x f x x F f xα αδ δ α= +

then Equation (49) yields

( ) ( ) ( ) ( )2 11 1 2 1 1

02 2 2 ; , .

nns s s k

t n n t t tk

r r r r r r kβ

δ δ δ δ−=

+ + + + + + + = ∑ r s s (50)

Comparing this equation with Equation (4.1) in [6], we get

( ) ( ); , , ; ,k s n k=r s s s α (51)

where ( )102 , 0,1, , 1ii m im r r i nα +== − + = −∑ and ( ), ; ,s n k sα are the generalized Comtet numbers of the first

B. S. El-Desouky et al.

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kind. Furthermore, using our notations, it is easy from Equation (4.4) in [6] and (41) to show that

( ) ( ) ( ) 0 1 1, ; , , ; , , , ,s

nk i

S n i n k S k i s s s s −=

= = + + +∑s r s sα (52)

where ( )102 , 0,1, , 1ii m im r r i nα +== − + = −∑ and ( ),S n k are the Stirling numbers of the second kind.

Next, we find a connection between ( ), ; ,n k r ss and the generalized harmonic numbers ( )inO which are de-

fined by, see [13] and [14],

( )

( )1

1 .2 1

ni

n ij

Oj=

=−

∑

From (42), we have

( )

( )( ) ( )( ) ( )( ) ( )( )

( )( ) ( )( ) ( )( ) ( )( )( )

( )( )

0

1 1 1 1

0 0 0 0

11 11 1

0 1 00 0

, ; ,

2 1 1 2 1 2 1 exp log 1 2 1

12 1 exp log 1 2 1 2 1 exp

2 1

2 1

nsk

k

n n n ns s s s

j j j j

i in nn ns s

ij i jj j

s

j

n k r s x

j r x j r j r x j r

xj r s x j r j r sij r

j r

=

− − − −

= = = =

+− −− ∞ −

= = == =

=

= + + + = + + +

− = + + + = + +

= +

∑

∏ ∏ ∏ ∏

∑ ∑ ∑∏ ∏

s

( ) ( ) ( )( ) ( ) ( )

( )( ) ( )( ) ( )

( )( ) ( )( ) ( )

1

1

1

1

1 11 1

1 0 10 0

1

00 1

1

0 1

1 1exp 2 1 !

2 1 1!

2 1 1 .!

l

l

l

l

li in n si ii in n ii

i l ij

iiln s k l kn nk

l k l i i kj l

iiln s k l kn nk

k l i i kj l

s O x j r s O x lr ir i

O Osj r xl i i r

O Osj r xl i i r

+ +− −∞ ∞ ∞

= = ==

− ∞ ∞+

= = + + ==

−+

= + + ==

− − = +

= + −

= + −

∑ ∑ ∑∏ ∏

∑ ∑ ∑∏

∑∏

0l

∞ ∞

=∑∑

Equating the coefficients of kx on both sides, we obtain

( ) ( ) ( )( ) ( )1

1

1

00 1

, ; , 2 1 1 .!

l

l

iilns k lns k n n

kl k l i i kj l

O Osn k r s r jl i i r

− ∞ ∞+−

= = + + ==

= + −∑ ∑ ∑∏

s (53)

From (22) and (53), we have the combinatorial identity

( ) ( ) ( )( ) ( )1

1

1

, 01 0 10

2 1 2 1 1 .!

lj

n lj

iiln ni s k l n n

kns k l k l i i kj jj l

i

s O Osj ji l i i rσ

− ∞ ∞+

= − = = + + == =≥

− = + −

∑ ∑ ∑ ∑∏ ∏

(54)

hence, setting 1s = , we get the identity

( ) ( ) ( )( ) ( )1

1

1

, 01 0 10

2 1 2 1 1 .!

lj

n lj

iin n ki k l n n

n k l i i kj j li

O Oj j

i i lσ

−+

= − = + + == =≥

− = + −

∑ ∑ ∑∏ ∏

(55)

4. Some Applications 4.1. Coherent State and Normal Ordering Coherent states play an important role in quantum mechanics especially in optics. The normally ordered form of the boson operator in which all the creation operators a+ stand to the left of the annihilation operators a. Using the properties of coherent states we can define and represent the generalized polynomial ( ),r sP x and generalized

B. S. El-Desouky et al.

1582

number ,r sP as follows. Definition 4.1 The generalized polynomial ( ),r sP x is defined by

( ) ( ),0

; , ,n

kr s

kP x k x

β

=

= ∑ r ss (56)

and the generalized number ,r sP

( ) ( ), ,0

1 ; , .n

r s r sk

P P kβ

=

= = ∑ r ss (57)

For convenience we apply the convention

( ); , 0 for 0 or .nk k k β= < >r ss (58)

Now we come back to normal ordering. Using the properties of coherent states, see [7], the coherent state ma-trix element of the boson string yields the generalized polynomial ( ),r sP x

( ) ( )( )( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

2 2 2 1 1 1

* *1 1 1

* *1 1

2 2 2

0 0 0

22 2

, , *1

e e e e e e

e ; , e ; , e , ,

e e , where .

n n n

n n nn nl l ll l l

n nl ll l

r a s r a r a s r a r a s r a

nr a r z r zk k k

k k k

nr z r zr s r s n j

j

z a a a z

z k a z k z z z k z

zP z P s

z

β β

β

+ + + + + +

+= = =

= =

= = =

=

∑ ∑ ∑

∑ ∑

= = =

= = =

∑ ∑ ∑

∑

r s r s r s

s s s

(59)

Definition 4.2 We define the polynomial ( ),P n x as

( ) ( )1

, , ,n

k

kP n x n k x

=

= ∑s (60)

and the numbers

( ) ( ) ( )1

,1 , .n

kP n P n n k

=

= = ∑s (61)

For convenience we apply the conventions

( ) ( ) ( ) ( ),0,0 and , 0 for and 0 0, 1.nn n k k n P P xδ= = > = =s s (62)

Similarly, using the properties of coherent states and (32) we have

( ) ( )

( ) ( )

( )

* *

* *

2

0

2 2

0 0

22 2

*

e e e ,

e , e ,

e , e , .

na a na k

k

n nnz k nz k

k k

nz nz

z a z z n k a z

n k z z z n k z

zP n z P n

z

+ + +

=

= =

=

= =

= =

∑

∑ ∑

s

s s (63)

4.2. Matrix Representation In this subsection we derive a matrix representation of some results obtained.

Let rs be n n× lower triangle matrix, where rs is the matrix whose entries are the numbers ( ), ; ,1n k rs ,

i.e. ( ), 0

, ; ,1 .r i ji j r

≥= s s Furthermore let rN be an n n× lower triangle matrix defined by

B. S. El-Desouky et al.

1583

( )2

, 0e , ; ,1irx

r i ji j r

≥ = N , rM is a diagonal matrix whose entries of the main diagonal are 2e , 0,1, ,irx i n= ,

i.e. ( ) ( ) ( ) ( )( )T0 10 2 4 2r rdiag e ,e ,e , ,e , e e , e e , , e e

nrx rx rx nrx rx rx rx rx rx rxD D D= =M R and

( )T0 1 2, , , , nD D D D D= .

Equation (27), may be represented in a matrix form as ,r r r r= =R N D M Ds (64)

for example if n = 3 then

( )( )( )( )

0

0 01

2 2 1

2 4 4 4 22

3 6 2 6 6 6 33

0

2

4

6

e ee 0 0 0

e e e e 0 03 e 4 e e 0e e

15 e 23 e 9 e ee e

1 0 0 0e 0 0 01 0 00 e 0 0

0 0 e 00 0 0 e

rx rx

rx

rx rx rx rx

rx rx rxrx rx

rx rx rx rx

rx rx

rx

rx

rx

rx

DD

D r Dr r DDr r r D

D

r

=

=

0

1

2 2

3 2 3

,3 4 1 0

15 23 9 1

DD

r r Dr r r D

(65)

its inverse is given by 1 1 1 .r r r r r− − −= =D N R M Rs (66)

Setting r = 1 in (64), we get

1 1 1 1 ,= =R N D M Ds (67)

( )( )( )( )

0

0 0 0 01

2 2 1 2 1

4 4 4 2 4 22

6 6 6 6 3 6 33

e e1 0 0 0e 0 0 0 e 0 0 0

e e 1 1 0 0e e 0 0 0 e 0 03 4 1 03e 4e e 0 0 0 e 0e e

15 23 9 115e 23e 9e e 0 0 0 ee e

x x

x x

x x x x x

x x x xx x

x x x x x

x x

DD D

D D DD DDD D

D

= =

,

(68)

hence 1 1 1

1 1 1 1 1.− − −= =D N R M Rs

For n = 3, we have

( )( )( )( )

( )0 0

0 01

21 2

2 42 42

2 4 63 63

e e e e1 0 0 0 1 0 0 0 e 0 0 0

e e1 e 0 0 1 1 0 0 0 e 0 01 4e e 0 1 4 1 0 0 0 e 0e e1 13e 9e e 1 13 9 1 0 0 0 e

e e

x x x x

x

x xx x

x x xx x

x x x x

x x

D DD

DDD DD

D

− −

− − −

− − − −

− − = = − − − − − −

( )( )( )

1

2

3

e e

e e

e e

x x

x x

x x

D

D

D

(69)

5. Conclusion In this article we investigated a new family of generalized Stirling numbers of the first kind. Recurrence rela-tions and an explicit formula of these numbers are derived. Moreover some interesting special cases and new

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1584

combinatorial identities are obtained. A connection between this family and the generalized harmonic numbers is given. Finally, some applications in coherent states and matrix representation of some results are obtained.

References [1] El-Desouky, B.S. (2011) Generalized String Numbers of the First Kind: Modified Approach. Journal of Pure and Ma-

thematics: Advances and Applications, 5, 43-59.

[2] Gould, H.W. (1964) The Operator ( )nxa ∆ and Stirling Numbers of the First Kind. The American Mathematical

Monthly, 71, 850-858. http://dx.doi.org/10.2307/2312391 [3] Comtet, L. (1972) Nombres de stirling généraux et fonctions symétriques. Comptes Rendus de l’Académie des Sciences

(Series A), 275, 747-750. [4] Comtet, L. (1974) Advanced Combinatorics: The Art of Finite and Infinite Expiations. D. Reidel Publishing Company,

Dordrecht, Holand. http://dx.doi.org/10.1007/978-94-010-2196-8 [5] Comtet, L. (1973) Une formule explicite pour les puissances successive de l’operateur derivation de Lie. Comptes

Rendus de l’Académie des Sciences (Series A), 276, 165-168. [6] El-Desouky, B.S. and Cakić, N.P. (2011) Generalized Higher Order Stirling Numbers. Mathematical and Computer

Modelling, 54, 2848-2857. http://dx.doi.org/10.1016/j.mcm.2011.07.005 [7] Blasiak, P. (2005) Combinatorics of Boson Normal Ordering and Some Applications. PhD Thesis, University of Paris,

Paris. http://arxiv.org/pdf/quant-ph/0507206.pdf [8] Blasiak, P., Penson, K.A. and Solomon, A.I. (2003) The General Boson Normal Ordering Problem. Physics Letters A,

309, 198-205. http://dx.doi.org/10.1016/S0375-9601(03)00194-4 [9] Cakic, N.P. (1980) On Some Combinatorial Identities. Applicable Analysis and Discrete Mathematics, 678-715, 91-94. [10] Carlitz, L. (1932) On Arrays of Numbers. American Journal of Mathematics, 54, 739-752.

http://dx.doi.org/10.2307/2371100 [11] El-Desouky, B.S., Cakic, N.P. and Mansour, T. (2010) Modified Approach to Generalized Stirling Numbers via Diffe-

rential Operators. Applied Mathematics Letters, 23, 115-120. http://dx.doi.org/10.1016/j.aml.2009.08.018 [12] Viskov, O.V. and Srivastava, H.M. (1994) New Approaches to Certain Identities Involving Differential Operators.

Journal of Mathematical Analysis and Applications, 186, 1-10. http://dx.doi.org/10.1006/jmaa.1994.1281 [13] Macdonald, I.G. (1979) Symmetric Functions and Hall Polynomials. Clarendon (Oxford University) Press, Oxford,

London and New York. [14] Choi, J. and Srivastava, H.M. (2011) Some Summation Formulas Involving Harmonic Numbers and Generalized Har-

monic Numbers. Mathematical and Computer Modelling, 54, 2220-2234. http://dx.doi.org/10.1016/j.mcm.2011.05.032.

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Appendix Tables of ( , ; , )n k r ss calculated using Maple, for some values of n, k, r and s:

Table 1. 0 ≤ n, k ≤ 4, r = s = 1.

1 0 0 0 0 1 1 1 0 0 0 2 3 4 1 0 0 8

15 23 9 1 0 48 105 176 86 16 1 384

Table 2. 0 ≤ n, k ≤ 4, r = s = 2.

1 0 0 0 0 0 0 0 0 1 4 4 1 0 0 0 0 0 0 9

144 192 88 16 1 0 0 0 0 441 14400 22080 12784 3552 508 36 1 0 0 53361

2822400 4730880 3138304 1076224 211808 24832 1712 64 1 12006225

Table 3. 0 ≤ n, k ≤ 4, r = 2, and s = 3.

1 0 0 0 0 0 0 0 0 0 0 0 1 8 12 6 1 0 0 0 0 0 0 0 0 27

1728 3456 2736 1088 228 24 1 0 0 0 0 0 9261 1728000 3974400 3824640 2014208 639936 127776 16128 1248 54 1 0 0 12326391

4741632000 11921817600 12904335360 7944527872 3104947968 515321088 148279040 18914304 1687152 103040 4104 961 41601569625

Notice that the last column in all tables is just the sum of the entries of the corresponding row.