2-Variable generalized hermite matrix polynomials and lie algebra representation

Vol. 66 (2010) REPORTS ON MATHEMATICAL PHYSICS No.2

2-VARIABLE GENERALIZED HERMITE MATRIX POLYNOMIALSAND LIE ALGEBRA REPRESENTATION**

SUBUHI KHAN l and NUSRAT RAZA

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India(e-mail: [email protected])

(Received September 14, 2009 - Revised May 28, 2010)

In this paper, we introduce the 2-variable generalized Hermite matrix polynomials(2VGHMaP) H;(x, y; A) and discuss their special properties. Further we use the representation theory of the harmonic oscillator Lie algebra 9(0, 1) to derive certain results involvingthese polynomials. Furthermore, as applications of our main results, we derive the generatingrelations for the ordinary as well as matrix polynomials related to 2VGHMaP H;(x, y; A).

Keywords: 2-variable generalized Hermite matrix polynomials, Lie group, Lie Algebra, representation theory, generating relations.

AMS Classifications: 33C50, 33C80.

1. Introduction

Special matrix functions appear in statistics, Lie group theory and number theory[3, 12, 19, 22]. In the last two decades matrix polynomials have become importantand some results of the classical orthogonal polynomials have been extended toorthogonal matrix polynomials, see for example [10, 11, 13, 16, 20]. Hermite matrixpolynomials have been introduced and studied in [14, 15] for matrices in cm x m

whose eigenvalues are all situated in the right open half-plane. Some properties ofHermite matrix polynomials are discussed in [7, 8].

If A is a matrix in cm x m , its spectrum a(A) denotes the set of all the eigenvaluesof A and a(A) = max{Re(z) : z E a(A)}, f3(A) = min{Re(z) : z E a(A)}. If fez)and g(z) are holomorphic functions in an open set Q of the complex plane andif a(A) c Q, then following the Riesz-Dunford functional calculus [9, p. 558],we denote by f(A) and g(A) the images of functions fez) and g(z), respectively,acting on the matrix A and

f(A)g(A) = g(A)f(A). (1.1)

If Do is the complex plane cut along the negative real axis and log(z) denotes theprincipal logarithm of z, then Zl/2 represents exp(110g(Z)). If the matrix A E cm x m

I Corresponding author.**This work has been done under the Major Research Project No. F.33-11012007 (SR) sanctioned to the first

author by the University Grants Commission, Government of India, New Delhi.

[159]

160 S. KHAN and N. RAZA

with a(A)c Do, then A1/2 = JA denotes the image by ZI/2 of the matrix functionalcalculus acting on the matrix A.

Let A be a matrix in cn x n such that

Re(/-L)i 0, /-L E a(A). (1.2)

Then the Hermite matrix polynomials Hn(x; A) are defined by [14, p. 25]

[n/2] (l)k

Hn(x; A) = n! L - (xJ2A)n-2k, n ::: 0 (1.3)k=O (n - 2k)! k!

polynomials

(1.5)

(1.4)

(1.7)

(1.6)

00 tn

exp(2xt - t2) = L Hn(x)-.

n=O n!

In view of Eqs. (1.4) and (1.5), we observe that

Hn(x; A) = Hn(xH).

Further, we recall that the 2-variable Hermite Kampe de Feriet(2VHKdFP) H;(x, y) [2] are defined by the generating function

DO t"exp(xt + yt2

) = L Hn(x, y)-n=O n!

and specified by the generating function

00 t"exp(xtJ2A - t

21)= LHn(X; A)"n=O n.

We recall that the generating function for ordinary Hermite polynomialsis given by [1]

and the series definition for 2VHKdFP H;(x, y) is given by [2]

, ~ xn-

2k lHn(x, y) = n. £:0 (n _ 2k)! k! . (1.8)

The 2VHKdFP H;(x, y) are related to the classical Hermite polynomials H;(x)or Hen(x) as

(1.9a)

and 1Hn(x, -2:) = Hen(x).

Also, there exists the following close relationship [2, p. 341(21)],

Hn(x, y) = (i,JYt tt, (~) = in(2y)~Hen (. ~),21~ ly2y

(1.9b)

(1.10)

(2.1)

(2.2)

2-VARIABLE GENERALIZED HERMITE MATRIX POLYNOMIALS AND LIE ALGEBRA... 161

with the classical Hermite polynomials. The usage of a second variable (parameter)y in the 2VHKdFP H; (x, y) is found to be convenient from the viewpoint oftheir applications. The importance of generalized Hermite polynomials has beenrecognized [4, 5] and these polynomials have been exploited to deal with quantummechanical and optical beam transport problems. The introduction of multi-variablespecial functions serves as an analytical foundation for the majority of problemsin mathematical physics that have been solved exactly and finds broad practicalapplications. For some physical problems the use of new classes of special functionsprovided solutions hardly achievable with conventional analytical and numericalmeans. Very recently the first author and Hassan [17] introduced the 2-variableforms of Laguerre and modified Laguerre matrix polynomials and framed them intothe context of the representations of Lie algebras.

Motivated by the work of Jodar and his co-authors on Hermite matrix polynomials,see for example [7, 8, 14, 15], and due to the importance of 2-variable forms ofHermite polynomials [2, 4, 5], we introduce in this paper the 2-variable generalizedHermite matrix polynomials and derive the generating relations involving thesepolynomials by using Lie algebraic techniques. In Section 2, we introduce the2-variable generalized Hermite matrix polynomials (2VGHMaP) Hj;(x, y; A) andestablish their generating function, series definition, differential equation and otherproperties. In Section 3, we use the representation theory of the Lie algebra g(O, 1)to derive generating relations involving 2VGHMaP Hj;(x, y; A). In Section 4, wederive the generating relations for the polynomials related to 2VGHMaP Hj;(x, y; A)as applications of our main results.

2. 2-variable generalized Hermite matrix polynomials

In view of Eqs. (1.4), (1.6) and (LlO), the generating function for 2-variableHermite matrix polynomials (2VHMaP) HII (x, y; A) can be cast in the form

( fI )00 til

exp xt _+yt2/ =LHn(x,y;A)l'

2 11=0 n.

where A is a matrix in cm x m .

In order to find the series definition for 2VHMaP HII (x, y; A), we consider thematrix valued function

GA(x, y; t) = exp(xtfI + yt 2/ ).

Further, in view of Eq. (2.1) and due to the fact that G A (x, y; t) is a functionof the complex variable t, we can represent GA(x, y; t) by a power series at t = °of the form

00 tilGA(x, y; t) = L n.«, y; A)-.

11=0 n!(2.3)


(2.4)

Now, due to the fact that the exponentials on the r.h.s. of Eq. (2.2) commute,we have

(fA)nXy 2" lt

n+

Zk

GA(A, y; t) = exp(xtH) exp(yt'/) = ~~ 2n!k! '

which on rearranging the series becomes

(fA)n-Zk

00 [n/Z] Xy 2" ltn

GA(X, y; t) = ~ {; (n _ 2k)!k!

Thus, from Eqs. (2.3), (2.4) and by identification of the coefficient of t", weobtain the following series definition for 2VHMaP Hn(x, y; A):

'./2, (xHf"lHn(x, y; A) = n! {; (n _ 2k)! k!' n 2: O. (2.5)

Recently, Sayyed et al. [20, p. 273 (2.1)] introduced the generalized Hermitematrix polynomials H;.m(x; A) as

00

exp()..(xtm - tmI) = L H;,m(x; A)tn,

n=O

where n = 0, 1,2, ..., AE ~+ and m is a positive integer.For our purpose, we consider the generalized Hermite matrix polynomials

H; m(x; A), for m = 2, i.e.

00 t"exp()..(xthA - tZI) = L H;(x; A)-. (2.6)

n=O n!

In view of Eqs. (1.4), (2.1) and (2.6), we introduce the 2-variable generalizedHermite matrix polynomials (2VGHMaP) H;(x, y; A) as

exp(J.(xtH + yt'/)) = ~ H;(x, y; A) ~. (2.7)

Following the same lines of proof as in the case of Hn(x, y; A), we find theseries definition for H;(x, y; A) as

(

fA)n-Zk k

[n/Z] xY"2 yHA(x y' A) - n,)..n~-"--_--,-__

n " -. f;;o )..k(n - 2k)!k!(2.8)

(2.9)

2-VARIABLE GENERALIZED HERMITE MATRIX POLYNOMIALS AND LIE ALGEBRA ... 163

For the sake of transparency, 2VGHMaP H:(x, y; A) for n ::: 3 correspondingto some physically and algebraically important matrices are listed below:

Ht(x, y; A) = I,

A fAH] (x, y; A) = Axy i'

H{(x, y; A) = A2X2( ~) + 2AyI,

H;(x, y; A) = A3(Xj%Y

+6A2(Xj%)y.

We observe that H:(x, y; A) and H:(x; A) are related as

H;(x, y; A) = (i./YtH; (2i~; A)and consequently Hn(x, y; A) and Hn(x, y) are related as

u.«. y; A) = Hn(Xj%, y).

(2.10)

(2.11)

Now, by differentiating the generating function (2.7) with respect to t, x and y,we get the following pure and differential matrix recurrence relations satisfied by2VGHMaP H:(x, y; A),

H;(x, y; A) = A(xj%HLt (x, y; A) + 2(n - l)yH;_2(x, y; A)) (2.12)

n ~ ],

and

Xj%DxH;(X, y; A) + 2nyDxHLl (x, y; A) - nj%H:(X, y; A) = 0,

A fAAxiH:-t (x, y; A) - YiH:(x, y; A) + 2yDxH:_t(x, y; A) = 0, n ~ 1,

xj%DY H; (x , y; A) + 2nyDyH;_1 (x, y; A) - n(n - I)H;_t (x, y; A) = 0,

2yDyH;(x, y; A) - n ( H;(x, y; A) - Axj%HLI (x, y; A)) = 0, n ~ 1,

respectively.We derive the k'" derivative of H:(x, y; A) with respect to x as

(2.13)

n ~ 1,

, (j%)kk A. n. A.DxHn(x, y, A) = (n _ k)! A 2" Hn_k(x, y, A), k ::: n, (2.]4)


and with respect to y as

k A n! ADyHn (x, y; A) = Hn - 2k(X, y; A),

(n - 2k)!(2.15)

From the recurrence relations (2.12) and (2.13), we conclude that 2VGHMaPH;(x, y; A) are the solutions of the following matrix differential equation,

n ~ O.( 4Y a22 + AxA~ - nAA) H~(x, y; A) = 0,ax ax

Further, we note the following special cases of 2VGHMaP H;(x, y; A):

I. For A= 1, we haveH~(x, y; A) = Hn(x, y; A),

(2.16)

(2.17)

where Hn(x, y; A) are 2VHMaP defined by Eq. (2.1). Again, in view of Eq. (2.11),for A = 2 Eel x I, we have

Hn(x, y; 2) = Hn(x, y),

where Hn(x, y) are the 2VHKdFP defined by Eq. (1.7).

II. For y = -1 and x -+ 2x, we have

H~(2x, -1; A) = H~(x; A), (2.18a)

where Hj;(x; A) are the GHMaP defined by Eq. (2.6). Again, for A = 1, we have

Hn(2x, -1; A) = Hn(x; A), (2.18b)

where Hn(x; A) are the HMaP defined by Eq. (1.4).

III. For A= I = 1, y = -1, A = 2 E C1x 1 and x -+ 2x, we have

H~ (2x, -1; 2) = Hn(x), (2.19)

where H; (x) are the Hermite polynomials.

IV. For y = 0, we have

and for x = 0 we have

(2.20a)

{

)..m(2m)! m

H;(O,y;A) = m! y,0,

if n = 2m,

if n=2m+1.(2.20b)


3. Representation of the Lie algebra Q(O, 1) and the generating relations

The irreducible representation t ai.]: of the harmonic oscillator Lie algebra yeO, 1)is defined for all w, u. E C such that j.), 1= 0. The spectrum of this representation isthe set S = {-w +n: n a nonnegative integer} and the representation space V hasa basis (fn)nES, such that

J3In = nln, Ef; = j.),ln, J+In = j.),ln+l,

J- In = (n +W)ln-l, Co,dn = (1+ J- - EJ3)ln = j.),wln,(3.1)

for all n E S. The operators J+, J-, J 3, E satisfy the following commutationrelations:

(3.2)

We can take to =°and u. = 1, without any loss of generality. For this choiceof wand j.)" we consider the irreducible representation to,1 of g(O, 1) with thespectrum as the set S = {n: n a nonnegative integer} and the representationspace V with the basis (fn)nES such that

J3In = nln,

J- In = nln-I,

Ef; = In, J+In = In+I'

Co,Jin = (1+ J- - EJ3)ln = 0,(3.3)

for all n E S, the commutation relations satisfied by the operators J+, J-, J 3, Eare the same as in Eq (3.2).

In particular, we are looking for the matrix functions In(x, y, t) = Zn(x, y; A):",where A is a matrix in (Cmxm, such that Eqs. (3.3) are satisfied for all n E S. Thereare numerous possible solutions of Eq. (3.2). We consider the linear differentialoperators J+, J-, J 3, E of the following form:

(3.4)

The operators in Eq. (3.4) satisfy the commutation relations (3.2). In terms ofthe functions Zn(x, y; A) and using operators (3.4), relations (3.3) become


(3.5)= nZn- 1(x , y; A),

(i)

(iii)

(ii)

(2Y(H)- I aax

+AxH)Zn(x , Y; A) = Zn+I (X , y; A),

1 ( fA)-1aJ: V2 ax Zn(X, y; A)

( 4Y a22 + AXA~ - nAA) z,«,Y; A) =0.ax ax

In view of Eqs. (2.12), (2.13) and (2.15), we observe that

Zn(x , y; A) = H: (x , y; A)

satisfy Eqs. (3.5).Thus we conclude that the matrix functions

!n(x , y; t ) = H:(x, y; A):", n E S,

form a basis for a realization of the representation to,1 of g(0, 1). This representationof g (O, 1) can be extended to a local multiplier representation of the Lie groupG (O, 1) , defined on F , the space of all function s analyt ic in a neighbourhood ofthe point (xo, yO, to) = 0 , 1, 1) .

Now, we proceed to compute the multipli er representation [T(g)!](x , y; t ),g E G (O, 1) induced by the i-operators (3.4). First we compute the action s ofexp(b,TT-), exp(c.1-), exp(r .13) and exp(a£ ) on lex , y; t) , where .1+, .1-, .13

and E are basis elements of the Lie algebra g(0, 1) [18, p. 10]. Using the result [18,p. 18 (Theorem 1.10)] and the operators (3.4), the local multiplier repre sentationtakes the form:

[T(expb.1+)f](x , y; t ) = exp(A(bxtfI + b2yt 2I) ) !(x + 2byt (fI) - I, y; t).

[T(exp c.T )f](x , y ;t)= f(X+ :t (m- I , y ; t). (3.6)

[T (expr.13)f](x, y; t) = l ex , y; teT),

[T(expa£)f](x, y; t) = exp(a )! (x , y; t),

for ! E :F.If g E g (O, 1) has parameters (a, b, c, r ), then

g = exp(b.1+) exp(c.1- ) exp(r .13) exp(a£ ).

Thus,

[T(g)f](x , y; t ) = [T(exp(b.1+»T(exp(c.1-»T(exp(r.13»T(exp(a£»f] (x , y; t),


and therefore we obtain

[T(g)f](x, y; t) = exp(aI +'A(bxt~ +b2Yt 21))

(

CI +Axt~+ 2'Abytl )x f ~2A ' y; te

r. (3.7)

At -2

The matrix elements of T(g) with respect to the analytic basis (fn)nES are thefunctions Bkn(g) uniquely determined by to.! of g(O, 1) and are defined by

(3.8)n = 0,1,2, ....00

[T(g)fn](x, y; t) = L Bkn(g)!k(x, y; t),k=O

Therefore, we prove the following result.

THEOREM 3.1. Thefollowing generatingequation involving2VGHMaP H;(x, y; A)holds,

00

= Lcn-kLin-k)(-be)Hl-(x, y; A)t k-n. (3.9)k=O

Proof: Using Eqs. (3.7) and (3.8), we obtain

00

= L Bkn(g)Ht(x, y; A)tk,

k=O

n=0,1,2, ... , (3.10)

and the matrix elements Bkn(g) are given by [18, p. 87 (4.26)]

Bkn(g)=exp(a+nr)en-kLt-k\-be), k,n~O. (3.11)

where Lin-k

) ( -be) denotes the associated Laguerre polynomials [1].Substituting the value of Bi; given by Eq. (3.11) into Eq. (3.10) and simplifying,

we obtain the result (3.9). 0


(3.12)

if n < 0,

if n ~ 0,

REMARK 3.1. Taking b ~ °in the generating function (3.9) and making useof the limit [18, p. 88 (4.29)]

enL?(be) Ib=O = { (: + l)en,

0,

we deduce the following consequence of Theorem 3.1.

COROLLARY 3.1. The following summation formula for 2VGHMaP H;(x, y; A)holds:

(3.14)if n ~ 0,

if n > 0,

(

cI + sxt fA ) n

H; Atd"2, Y; A ~ ~c"-'C ~ k)Ht(X, Y; A)tk- " . (3.13)

REMARK 3.2. Taking e ~ °in the generating function (3.9) and making useof the limit [18, p. 88 (4.29)]

enL?(be) Ic=o = {~:"'W",(-n)!

we deduce the following consequence of Theorem 3.1.

COROLLARY 3.2. The following generating equation involving 2VGHMaPH;(x, y; A) holds:

eXP(AbxtH + Ab2yt2/)H;(X/ + 2bYt(H)-1, y; A)CXl bk- n

= "" HA(x y' A)tk-

n• (3.15)

~ (k - n)! k ' ,

REMARK 3.3. Taking Y = -1 and replacing x by 2x in Eq. (3.9) and thenmaking use of Eq. (2.17(a)), we deduce the following consequence of Theorem 3.1.

COROLLARY 3.3. The following generating equation involving GHMaP H;(x; A)holds,

CXl

= Len-kLt-k\-be)Ht(x; A)tk- n. (3.16)

k=O


Further, taking b -+ 0 in the generating relation (3.16) and making use of thelimit (3.12), we find the following summation formula for GHMaP H;(x; A),

H: (XI + :t (hA)-I; A) = t,cn-k(n ~ k)H;(X; A)tk-n. (3.17)

Again, taking c -+ 0 in the generating function (3.16) and making use of thelimit (3.14), we find the following generating relation involving GHMaP H;(x; A),

00 bk- n

exp(AbxtJ2A - Ab2t2I)H;(xI - 2bt(J2A)-I; A) = L H;(x; A)l-n.k=n (k - n)!

(3.18)

4. Special cases

We consider some special cases of Eqs. (3.9), (3.13), (3.15) and (3.16).

1. Taking A= 1 in Eqs. (3.9), (3.13) and (3.15) and using Eq. (2.16), we get thefollowing results involving 2VHMaP Hn(x, y; A):

( fA )A cI + xtv 2: + 2byt2I

eXP(bxtfI +b'yt'J )H' tk 'y; A

00

= Lcn-kLin-k\-bc)Hk(X, y; A)tk-n, (4.1)k=O

and

(4.3)respectively.

II. Taking A = I = 1 and A = 2 E C 1x l in Eq. (3.9), we get the following generatingrelation involving 2VHKdFP Hn(x, y),

(c + xt + 2byt

2) ~ (n kexp(bxt+b2yt2)H

n ,y = i-Jcn-kLk - \-bc)Hk(x,y)tk-n, (4.4)t k=O

which for t = 1, yields [6, p. 84(21)].

170 s. KHAN and N. RAZA

For the same choice of A, I and A, Eq. (3.13) gives the following summationformula involving 2VHKdFP Hn(x, y),

n, (x +~, y) = ten-k( ~ )Hk(X, y)tk- n, (4.5)t k=O n k

which for t = 1, yields [6, p. 87(48)].Again for the same choice of A, I and A, Eq. (3.15) gives the following

generating relation involving 2VHKdFP Hn(x, y),

00 bk- n

exp(bxt + b2yt2)Hn(x + 2byt, y) = L Hk(x, y)tk-n, (4.6)

k=n (k-n)!

which for t = 1 or b = 1, yields [6, p. 87(46)].

III. Taking A= 1 in Eq. (3.16), we get the following generating relation involvingHMaP Hn(x; A),

(cI +xtm - 2bt

2I )exp(bxtm - b

2PI)Hn tm ; A

00

= L en- kL~n-k) (-be)Hk(x; A)tk-n. (4.7)

k=O

Further, taking A = 2 E Cl x I, I = 1 E Cl x I and t ---+ -t, we obtain

exp( -2bxt - b2t2)Hn (x + bt - ~) = f en- kL~n-kJc-be)u; (x)(-tr: (4.8)

2t k=O

which is a result of Miller [18, p. 106 (4.76)] and for t = -1, it yields [6, p. 87(43)].

Again, taking X = 1 in Eq. (3.17), we get the following summation formulainvolving HMaP Hn(x; A),

Hn(X+~(m)-I;A)= ten-k( ~k)Hk(X;A)tk-n, (4.9)t ~o n

which for A = 2 E Cl x l , I = 1 E Cl x l and t = 1 yields [18, p. 106]

n, (x +~) = ten-k( ~ ) Hk(X)t k-n. (4.10)2 k=O n k

Further, taking ).. = 1 in Eq. (3.18), we get the following generating relationinvolving HMaP Hn(x; A),

2-VARIABLE GENERALIZED HERMITE MATRIX POLYNOMIALS AND LIE ALGEBRA... 171

which for A = 2 E C1x 1, 1=1 E C1x 1 and t = -1 yields [18, p. 106]

00 (_b)k-nexp(-2bx - b2)H

n(x + b) = L Hdx).k=n (k - n)!

(4.12)

In this paper, we have introduced the 2-variable generalized Hermite polynomialsand framed these polynomials into the context of Lie algebra representation. Theanalysis presented in this paper confirms the possibility of introducing multi-variableforms of other matrix polynomials and functions. However, the merging of thesepolynomials with Lie algebraic techniques is also interesting for further researchwork.

5. Appendix

(5.1)if n = 2m + 1.

if n = 2m,

The Hermite polynomials and functions and their relations to other specialfunctions like the Laguerre ones determine a domain of great importance formathematical analysis and quantum mechanics. There are different ways of introducingthese polynomials, some of which are the methods of generating functions, relationsof power series coefficients, linear differential equations, Volterra integral equations.In this paper the first three methods are used to introduce 2VGHMaP H:(x, y; A).Since the last one is also important in view of the uniqueness and provides ananalytic algorithm related to physical background, we therefore derive here theVolterra integral equation of 2VGHMaP H:(x, y; A). In view of Eqs. (2.14) and(2.20b), we have

a 10

';- (H;(x, y; A)) I = Am+1(2m + 1)' Hox x=o " v" -2'

m.

(5.2)n ~ O.

Moreover, Eq. (2.16) can also be expressed as

(a2 hA a nAA) A-+---- H (x,y;A)=O,

ax2 4y ax 4y n

Now we deal with the problem of obtaining the integral equation from the abovedifferential equation along with the initial conditions, given by Eqs. (2.20b) and(5.1). First, we consider the case, when n is even, i.e. n = 2m, then Eq. (5.2)becomes

m ~ O.(a2 AxA a 2mAA) A-2 +--, - -- HZm(X' y; A) = 0,ax 4y ax 4y

From Eq. (2.14), we have

a2 (A 2A)axz H;m(x, y; A) = (2m)(2m - 1) 2 H;m_2(X, y; A).

(5.3)

(5.4)


(5.5)

Integrating Eq. (5.4) and using the initial conditions , given by Eqs. (2.20b) and(5.1), we get

a (A2A)l X

-H{m(x , y ; A) = (2m)(2m - 1) - H{m-2(~ ' y; A)d~ax 2 0

and

(>-?A) l X

Am

(2m) !H{m(x, y; A) = (2m)(2m - 1) - (x - ~)H{m-2(~ ' y ; A)d~ + ym.2 0 m !

(5.6)

In view of Eqs. (5.4)-(5.6), the differential equation (5.3) reduces to

4yH{m_2(X, y; A) - AAlx

{(2m - l)x - (2m)~}H{m_2(~' y; A)d~

Am-I (2m - 2)!- 4 ym = 0. (5.7)

(m - 1)!

Replacing m by m + 1 in Eq. (5.7), we get

lx Am (2m)'

4yH{m(x, y; A) - AA {(2m+ 1)x- (2m+2)~}H{m(~' y ; A)d~ -4 .ym+l = 0.o (m)!

(5.8)

m ::: 0.

m ~ 0, (5.10)

Next we consider the differential equation (5.2) , for n = 2m + 1, i.e.

(a2 AxA a (2m + I)AA) ).

ax2 + 4Y ax - 4y H 2m+1(x, y; A) = 0,

Following the same arguments, differential equation (5.9) reduces to

4yH{m_1 (x, y; A) - AAl x

{2mx - (2m + I)~}Him-l (~ , y; A)d~

Am-1

(2m -1)! m( H)-4 y Ax - = 0,(m - I)! 2

which on replacing m by m + I becomes

4yHt,,+1 (x, y ; A) - AAl x

{(2m + 2)x - (2m + 3)nH{m+l (~, y ; A)d~

Am (2m + I)! m+l ( H)-4 y Ax - m ~ 0.(m)! 2 '

(5.9)

(5.11 )

Finally, combining Eqs. (5.8) and (5.11), we get the Volterra integral equationof 2VGHMaP H:(x, y; A) in the form

2-VARIABLE GENERALIZED HERMITE MATRIX POLYNOMIALS AND LIE ALGEBRA. . . 173

4yHn\ x , y; A) - AAlx

(n + l)x - (n + 2)~}H:(~ , y; A)d~

A[n/ 2] , ( ~)n-2r_ 4 n. y [n/2]+ 1 AX - = 0([nI2]) ! 2 '

where r := [nI2] and [.] denotes the greatest integer function.

(5.12)

Acknowledgement

The authors are thankful to anonymous referee for several useful comments andsuggestions towards the improvement of this paper.

REFERENCES

[I] L. C. Andrews: Special fun ctions for Engineers and Applied Mathematicians, Macmillan PublishingCompany, New York 1985.

[2] P. Appell and 1. Kampe de Feriet: Functions Hypergeometriques et Hyperspheriques: Polynomes d'Hermite,Gauthier-Villars, Paris,1926.

[3] A. G. Constantine and R. J. Muirhead: Partial differential equations for hypergeometric functions of twoargument matrix, J. Multivariate Anal. 3 (1972) , 332-338 .

[4] G. Dattoli, C. Chiccoli, S. Lorenzutta, G. Maino and A. Torre: Generalized Bessel functions and generalizedHermite polynomials, J. Math. Anal. Appl. 178 (1993) , 509-516.

[5] G. Dattoli, S. Lorenzutta, G. Maino, A. Torre and C. Cesarano: Generalized Hermite polynomials andsuper-Gaussian forms, J. Math. Anal. Appl. 203 ( 1996), 597-609.

[6] G. Dattoli, H. M. Srivastava, and Subuhi Khan: Operational versus Lie-algebraic methods and the theoryof multi-variable Hermite polynomials, Integral Transf orms Spec. Funct. 16 (I) (2005), 81-91.

[7] E. Defez, M. Garcia-Honrubia and R. 1. Villanueva: A procedure for computing the exponential of a matrixusing Hermite matrix polynomials, Far East 1. Applied Mathematics 6 (3) (2002 ), 217-23 1.

[8] E. Defez and L. J6dar: Some applications of the Hermite matrix polynomials series expansions, J. CompoAppl. Math. 99 (1998), 105-117 .

[9] N. Dunford and J. Schwartz: Linear Operators, Vol. I, Interscience, New York 1957.[10] A. J. Duran: Markov's Theorem for orthogonal matrix polynomials, Can. J. Math. 48 (1996), 1180-1 095 .[I I] A. 1. Duran and W. Van Assche: Orthogonal matrix polynomials and higher order recurrence relations,

Linear Algebra Appl. 219 (1995), 26 1-280.[12] A. T. James: Special functions of matrix and single argument in statistics, in: Theory and Applications

of Special Functions, R.A. Askey (Ed.), Academic Press, New York (1975), p. 497-520.

[13] L. J6dar, R. Company and E. Navarro: Laguerre matrix polynomials and systems of second orderdifferential equations, Appl . Numer. Math. 15 (1994), 53-63.

[14] L. J6dar and R. Company: Hermite matrix polynomials and second order matrix differential equations,J. Approx. Theory App!. 12 (2) (1996), 20-30.

[15] L. J6dar, and E. Defez: On Hermite matrix polynomials and Hermite matrix function, J. Approx. TheoryAppl. 14 (1) (1998), 36-48.

[16] L. J6dar and J. Sastre: The growth of Laguerre matrix polynomials on bounded intervals. Appl. Math.Lett. 13 (2000), 2 1-26.

[17] Subuhi Khan and N. A. M. Hassan: 2-variable Laguerre matrix polynomials and Lie-algebraic techniques,J. Phys. A: Math. Theor. 43 (2010), 235204 (2 1 pp).

[18] W. Jr. Miller: Lie Theory and Special Functions, Academic Press, New York and London 1968.

[19] R. J. Muirhead: Systems of partial differential equations for hypergeometric functions of matrix argument,Ann. Math. Statist. 41 (1970), 99 1- 1001.


(20) K. A. M. Sayyed, M. S. Metwally and R. S. Batahan: On generalized Hermite matrix polynomials,Electronic Journal of Linear Algebra 10 (2003), 272-279.

(21) A. Sinap and W. Van Assche: Orthogonal matrix polynomials and applications, 1. CompoAppl. Math. 66(1996), 27- 52.

[22] A. Terras: Special functions for the symmetric space of positive matrices, SIAM J. Math. Anal . 16 (1985),620-640.

2-Variable generalized hermite matrix polynomials and lie algebra representation

Documents

Transcript of 2-Variable generalized hermite matrix polynomials and lie algebra representation