ON MODELS OF ^-REPRESENTATIONS OF sl(2, C) AND (/-CONFLUENT HYPERGEOMETRIC FUNCTIONS. V lv E K SAH AI AND SHALINI SRIVASTAVA. (Received October 2001)

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 33 (2004), ON MODELS OF ^-REPRESENTATIONS OF sl(2, C) AND (/-CONFLUENT HYPERGEOMETRIC FUNCTIONS V lv E K SAH AI AND SHALINI SRIVASTAVA (Received October 2001) Abstract. W e construct new two variable models of irreducible ^-representations of the Lie algebra sl(2, (D). The q-mellin transformaton is defined and used to transform the models in the form of ^-difference dilation models, involving basic hypergeometric function 2<t>i as basis functions. All the models culminate in many special function identities and recurrence relations. 1. Introduction Manocha [6] introduced the idea of irreducible (/-representations of the special complex Lie algebra si (2, C) and developed theory centering around it. A (/-analogue of fractional calculus technique [1, 7] was used to construct models of irreducible (/-representations of si (2, C) followed by the Weisner s method [18] to obtain identities involving (/-special functions of one and several variables. These models were in terms of (/-differintegral operators acting on (/-hypergeometric functions 20i- Further, in [15], (/-differintegral operator models of sl(2,c) with basis functions involving k2+i(f>k2 functions were constructed and a good number of identities were derived from them. The theory of ^-representation of Lie algebras together with the theory of q- integral transformations has also been a rich source of results in (/-special function theory. In Sahai [12, 13, 14], the (/-Euler integral transformation, which is motivated by (/-integral representation of beta function, is utilized to obtain models in terms of (/-difference dilation operators of si (2, C) along with identities involving (/-special functions. Both these models, viz. (/-differintegral operator models and (/-difference dilation operator models, are of (/-representations Dq (u, a) and (u) of si (2, C) in the particular case u = 0. In this paper, we construct new models of (/-representations Dq (u, a) and f q (u) of si (2, C) corresponding to u ^ 0 acting on (/-confluent hypergeometric functions i< i. We then define g-mellin transformation, which is motivated by (/-integral representation of gamma function. The (/-Mellin transformation is the ^-analogue of Mellin integral transformation used in [5, 10]. We utilize this transformation to transform these models of si (2, C) to new models involving (/-difference dilation operators with the (/-hypergeometric functions 20i as basis functions. The discussion culminates in interesting identities and recurrence relations involving i 0i and 20i functions. Section-wise treatment is as follows A M S Mathematics Subject Classification: 33D80, 17B37.

2 166 VIVEK SAHAI AND SHALINI SRIVASTAVA In Section 2, we give definitions involving hypergeometric and basic hypergeometric series, needed for our discussion. In Section 3, Lie algebra si (2, C) is presented along with its g-deformed version. In Section 4, we construct model of Dq ( 7 / 2, a) and f q ( 7 / 2) in two variables acting on i<^i and obtain identities and recurrence relations based on it. In Section 5, we introduce g-mellin transformation and then utilize it to transform the models given in Section 4 to ^-difference dilation models acting on basis functions 2<f>i- The transformed models are further used in obtaining identities and recurrence relations involving 20i- Finally, in Section 6, we give an application of these techniques to g-laguerre polynomials. 2. Preliminaries The generalized basic or (/-hypergeometric series r0s is defined as [3] P*.( ;,,«) =IV r V w S \ O i...,b s J A^ (b 1:q)ri---(bs:q)n (q:q)n. n=0 where g-shifted factorial (a; q)n is defined by (2.1) (2-2) In other words, ( (1 - a) (1 - aq) (l - aq -1), n = 1,2,... ( a'^)n ~ { 1; n==0 (2-3) [ [(1 - aq-1) (l - aq~2) (1 - aq~n)} ; n = - l, - 2,.... The series r(f>s terminates if one of the numerator parameter is of the form q~m, m = 0,1,2,..., and q ^ 0. When 0 < \q\ < 1, the series r0s converges absolutely for all x if r < s; and for ar < 1 if r = s + 1. If \q\ > 1 and jc <, then also r<f)s converges absolutely. It diverges for x / 0 if 0 < \q\ < 1 and r > s + 1, and if \q\ > 1 and a; > j ~ ~ ^, unless it terminates. The generalized hypergeometric series rfs is defined by [9] \ _ ~ (a i)n (qr)n r " { A,..., A ) ^ 0 ( A ) n '" ( /3. )»! { ' where {a)n is Pochhammer s symbol defined by (a) = I a (a + 1)--'(a + n ~ 1) = ^ 5 n = 1,2,... V,n \ 1; n = 0. K J The g-analogue of the binomial function is " ) (X'iV oc The ^-analogues of the exponential functions are. \ = (ffm joo ; 9 < 1; x < 1 (2.6)

3 ^-REPRESENTATIONS OF sl(2,c) 167 and E q 0*0 = Y j Q(n. nv = kl < 1- (2-8) n= 0 'n We shall also make use of the function r, ( «) = SL^ r ( 1 ~ 9 ) 1~ (2-9) eq \Q) defined for a ^ 0,-1, 2,.... This is a g-analogue of the gamma function and satisfies the functional equation r, ( a + i) = ^ r, ( a ). (2.10) We need the g-analogue of Kampe de Feriet function [16] C'E E1 {(-,m m -* v) n% (7 j)m+n n f=1 (5j)m n f=1 ( * ') m\n\ in the form [6] (2.11) ±a -.B\B' ( (a ) : (& );(& '). x,y ^:E;E' { (c) : (c) ; (c') qrrf Y ' n/=l ( ^ L - h, 1':f=l fe; Q)m nf=l (fy <?) ^ n f=1(e:);g)mn f=: i (e$;g)n (g;g)m(g;g)n For r = s = 0, we denote the l.h.s. of (2.12) simply by,a:s;jb' / (a) : (b);(b') \ 9c:E-,E ^ (c) (e) ; (e;) x,2/y The ^-derivative operator is defined by a, ( / ( * ) ) = (1_ 9)a <2-13> where the ^-dilation operator Tx is given by The (/-integral [3] is defined by Tx (/(* )) = /(«*) (2-14) oo J f(t)dqt = (1-9) / (? n)9n. (2.15)

4 168 VIVEK SAHAI AND SHALINI SRIVASTAVA 3. Irreducible q representations o f sl( 2, C) The complex Lie algebra sl(2,c) = L{SL(2,C )}, the Lie algebra of complex Lie group SL (2, C) = a c b d : a, 6, c, d G C, ad be = 1 (3.1) consists of all 2 x 2 matrices with trace zero, that is, It has a basis J + = sl(2,c) = satisfying the commutation relations 2 / e (3.2) (3.3) = ± j ±, = 2 J. (3.4) Let Vq be a complex vector space consisting of g-special functions with a basis {4>x A G S'} such that the functions {f\ = limq_>i <f>\ \A S } form a basis of a vector space, say V. Let A(Vq) be the associative algebra of all linear operators on Vq over the complex field. A ^-representation of si (2, C) on Vq is a mapping pq : si (2, C) >A (Vq) satisfying the following conditions: (i) pq (ax + by) = apq (x) + bpq (y) (ii) There exists a Lie algebra representation p of sl(2) on V such that lim pq (x) (f)x = p[x)f\, q-> 1 for all x, y G sl(2, C) and a, b G C. The representation pq of sl(2, C) is said to be irreducible if there is no proper subspace Wq of Vq which is invariant under pq. Define J f = P q(j+), J < = P q(j~), J q =P q(j0), (3.5) where J+, J~,./ S A(Vq). Manocha [6] has defined the following commutator rules for J^-operators: J qj+-qj+j t = J+, q jp ~ - J~ J q = -J ~, q J + J --J ~ J + = 292 J, - ( 1-9 ) 92" ^ J, u e C. (3.6) These commutator rules were later generalized by Sahai [11] for the 4-dimensional complex Lie algebra Q (a, b), a, b G C. Further, the commutation relations (3.6) are also equivalent to those introduced by Jimbo [4]. For details, see [6, 14, 15]. If we define an operator Cq on Vq by c q = qjtj7 q q q2ujq (3.7)

5 ^-REPRESENTATIONS OF sl(2,c) 169 then it is easy to check that (3.8) As q >1, the operators Jq, J~, Jq reduce to J +, J~, J which satisfies the commutation relations obeyed by si (2, C) and the operator Cq reduces to the Casimir operator C. Hence, as q * 1, pq reduces to a Lie algebra representation p of sl(2,c). For more details, see [6, 8]. 4. Models of Irreducible q representations We give below new two variable models for each of the ^-representations Dq («, a) and (u) corresponding to u = 7 / 2, in which case they will be denoted by Dq ( 7 / 2, 0;) and ]q ( 7 / 2), respectively Representation D q ( 7 / 2, a). Consider an irreducible representation Dq ( 7 / 2, a) of si (2, C), a, 7 G C and 7 ^ 0,-1, 2,... on the representation space Vq with basis {f\ A = a + n, n G Z }, such that action of si (2, C) on Vq is given by qx-< h. (4.1) To find a realization of (4.1) in terms of (/-dilation operators, we choose j ; = T- 1 (g - Tt - % - q~< ztztt), with basis functions 0 _ Q 1 A = a, a i l, a ± 2,.... (4.2)

6 170 VIVEK SAHAI AND SHALINI SRIVASTAVA Model (4.2) obeys (4.1) and the following J q J q Q Jq J q = ^, qj qj - - J - J aq = -J ~, q J +J --J -J + = 2<TV - (1 - q)q-'lj q Jj, 7 SC as well as qj^c, = cqj+, j - c, qcqjq JqC, C T Q g ' Identities based on model (4-2). As satisfies It immediately follows that fx cz,t) = 101 ^ r _ (1 - < T ^ ) (1 -.. (.. < ). * g : ( 1-9 ) f * (a; «) (*»';«) (6"; «), / \. a where ga = a, also satisfies ( 1-9 ) Using the fact that Cq (qsjq ) Cq Cqeq (s ), which follows from (4.4), we have c, [,. K - ),] 7-1(1-,).9J q> v /1:0;2 f a : ;b,b z,qt \. X 07-c' > 1 I (^?) where

7 ^-REPRESENTATIONS OF sl(2,<c) 171 Using the expansion OO [eq (sj*)u](z,t) = i n fa+n(z,t), (4.10) n 0 where in are obtained by putting 2 = 0, we get the following identity, after suitable rescaling: ( f f ;gl,i:q;2 f a--,b',b" z,-t / W Mill q, 1 ^ (o;«) /,A n a ( 1 qn+~'\ A ( aqn \ = 1 j ( ^ (? *) 301 ( c' J 1^ ^ (4'U) where it; = l-q Recurrence relations from model (4-2). The recurrence relations originating from the action of J+, J~ on f\ are given by (l - q x) (A + l\z) + qx1(j) 1 (\\qz) = i(j>i(\;z), (4.12) ' 1^1 (A;<jz) + 1^1 (\\qz) - (A;z) = (1 - qx~~l) i^i (A - l;gz) (4.13) respectively, where A (A; *) stands for ^ 4.2. R epresentation q ( 7 / 2 ). We look for an irreducible representation 9 ( 7 / 2) of sl(2,c), 7 G C {0, 1, 2,... }, on the representation space Vq with basis functions {/a I A = 0,1,2,... } such that the action of si (2, C) on Vq is given by J +h = 7 _ nx fx+1, 1-9 _ 1 0^ Jq fx = - ~ j fx-1, q 1 - q JT qfx = /A, q 9 l 1 -q (yjq Jq + 9 j? - 9 % ) A = 7/2+1^ 9A/a- (4.14) (1-9)

8 172 VIVEK SAHAI AND SHALINI SRIVASTAVA To meet this requirement, we choose J q = y ( < r 7 - J y r, - z 7 y r t ) Jq = r - ' m - r, ), ro 1 ~ 17/2Tt q ~ 1-9 C, = qjp~ +q~'<j0qj0q - q - J0q, (4.15) with basis functions h = 1^1 ( ;9 > -9 A+7^ * A> A = 0,1,2,... Model (4.15) satisfies (4.3) as well as (4.4) Identities based on model (4-15). As f\ = i0 i ^ ql ; g, - g A+7^ * \ satisfies it immediately follows that C,A (*, t) = ^ (2i t)) ( 1-9 ) (4.16)» (* -«) = -I;?;:; ) satisfies = E 7 7 (V' 101 ( < ;9 - «" +72) ±^(c';<?)n (g;<z)n V 7 n=0 v ^/n (4.17) Using the fact that c qu{z,t) = ( 1-9 ) (4.18) E, ( i SJ ) C, = C,, K " ), (4.19) which follows from (4.4), we have C [Eq (sjq ) u] (z, t) = (1 - <r7/2) (1 - q - r ^ 1) ( 1 - q f,w l ( (1=5)1. -q 1+xzt,qt 01:1:0 V «2.1, (4.20)

9 ^-REPRESENTATIONS OF sl(2,c) 173 where [Eq (sj -) u] (z, t) = ( : J f y l ; ). (4.21) The expansion leads to the following identity /i:0;i ( a : ; w/t -q*zt,t \ { d : </7 ; q\ l ) OO [Eq (Sj~ ) U] (z, t) = ^ An fn (z, t), (4.22) 71=0 = S d E f c k ( s ;< H ^ ( r ( ) Recurrence relations. The recurrence relations arising from model (4.15) are given by 101 (A;*) - ga+7i0 i (\\qz) -?A+72i0i (\;qz) = (l - qx+1) i0 i (A + 1;z), (4.24) i(f)i (\;qz) - qxi(j)i (A;z) = (l - qx) i0 i (A - 1;qz), (4.25) where i<j>i (A;z) stands for i4>i (. ;q, qx+7z 5. Transformed Models of si (2, C) We introduce the (/-Mellin transformation Iq. Define 13(13-1) 00 h (/?, x) = Iq [f (wz)] = ^ - - y J eq (-u) u0 V (ux) dqu, (5.1) q o which is motivated from the (/-integral representation of gamma function, as in [17], OO r q (7 ) = J eq (-it) u7_1 dqu. (5.2) 0 Using the substitution 2 = ux, which gives zaz uau = xax and Tzf (z) = f (qz) = Tuf (ux), we have the following transforms of certain operator expressions which are needed for the discussion. Iq [uf (ux)] = Eph ((3, x ), where Eph (/?, x) = h ((5 + 1, x). Iq[uAuf(ux)\ = xaxh (P, x), (5.3)

10 174 VIVEK SAHAI AND SHALINI SRIVASTAVA To obtain transformed models of Dq ( 7 / 2, 0;) and q ( 7 / 2), we put z = ux in models (4.2) and (4.15). Note that if pq is an irreducible representation of sl(2,c) on the representation space Vq having the basis vectors {f\ A S} in terms of {J+,J~,Jq}, then the transformation Iq induces another irreducible representation crq of sl(2,c ) on the representation space Wq IqVq in terms of { if+ = IqJ+Iq l,k q = IgJ-I-'tK = Iqjjjl-1} with basis vectors {hx = I h I A S} Transformed Model of D q ( 7 / 2, 0:). K i = r r ^ (1~ TxTt)' K~ = ^ T - 1 (q->tt ~Tx- ^ - ^ E l3q-''xtxtt K = 1 Zf ^ i. (5-4) 1 - q with basis functions as Indeed, h\ [x,t) = 20i ( Q Q~ ;q 97 ( 1-9 )? ' 5 K qk + -q K + K q = K+, gk qk q - K q K q = -K ~, q K + K ~ -K ~ K + = 2q~1K q (I q) q~yk qk q (5.5) and K+hx = q->' ^ h x+l, q 1 ~Q 1 aa-7/ 2 K q h \ = ^ h>, q 1 -Q Cqh\ = {qk+kq +q->k qk 0q -q-'>k q)hx Moreover, - < - - < - 7 (...) (! - q) qk+c'q = C'qK+, K~C'q = qc'qk q, K qc'q = CqK q. (5.7)

11 ^-REPRESENTATIONS OF sl(2,c) Identities based on transformed model >g ( 7/2, a). As is a solution of the function (*, t) = (! ~ g 7/2) ( i ~ 7/2+1) qx -,hx (X] t) _ (5.8) ( 1-9 ) w (x,t) = ( aj q^ c>b ^ l) ta is also a solution of S (c';g)n (g;g)n V? ( i - 9 ) r / C > (x,t ) = 7/2^ 1 ~2g 7/2+1V - 7 t»(*,t)- (5.10) ( 1-9 ) Using qk+cq C'qK +, we have C' [eq [skq ) u] (z, i) = ( ^ f r ^ ; g ) ^,- > (1 - g - ^ ) (1 - r - ' W ) ( 1 9)2 where x o;;!;? ( W, (5.1 1 ) V <r»-1(i-9) 9 1 ~ ^ / r_ (~ts+\ -.1 a:q0 ]b',b" q?x /c 1oN [eqr Jllj (x, t) ~ f st ^ ^ ast. ^7. ^ 1^ ^ 51J t. (5.12) From the expansion 59 L OO \eq (skq ) wj (x, i) = ^ ^A.nha-\.n (x, ), where An is found by putting x = 0, we arrive at the following identity after suitable rescaling: n=0 ( ^ ;9)oo di W a:<f-,v,w \ \q^ J OO ;1 v # : <f-y 1 ) (l-q)qf>, (5.13)

12 176 VIVEK SAHAI AND SHALINI SRIVASTAVA Recurrence relations from model (5.4)- The recurrence relations originating from the action of K+ and K~ on h\ are given by 20i (A;/?;x) - qx2(f>i {\-,(3;qx) = (l - qx) 2<t>i (A + 1; /?; x ), (5.14) xga-7- /? l _ J L 20x (A;/3 + 1 ; gx) + 20i (A; /?; gx) - qx~^20i (A; /?; x) qx qp \ where 20i (A; /?; x) stands for 20i ( ; q, \T-^ )q* ) = (1-9A_7) 20i (A - 1; /?; qx), (5.15) 5.2. Transformed Model of tq ( 7 / 2). ( «-7 - TxT, - E d xtxtt\, (5.16) k - = ^ r - 1(*A I - t A t), 1 - «7/2T( with basis functions as o~x ap n^+i-p \ h\ (x, t) = 201 ( ql ; q, y _ ~ ^ ) 1 Model (5.16) satisfies (5.5) and (5.7) and Kqh\ = K hx = q - T - q \ l - q kx < -e 'to.» rh 1 1 _ qa + 7 /2 K qhx = ^ fca, 1 - g C'qhx = (qk +K -+q~' (1 - g-7/2) (1 (1-9 qxhx. (5.17)

13 ^-REPRESENTATIONS OF sl(2,c) Identities based on transformed model t q ( 7 / 2). As satisfies / 0 -A a(3 QX + l-0 \ h\{x,t) = 201 f ' ql -,q, x j t, A = 0,1,2, This, in turn, gives C'qh\ (x,t) = (1 ~ Q 7/2^ 1 ~29 q*hx (x,t). (5.18) (! - 9 ) satisfies Using,.x /1:1 ;o ( a : q13; u(x,t) = 0 1:1O, ; 1-9 \ c. q ', n=0 h v (c';q)n /n («;?) v^ ^/n?7,9, 1-1 ) 1 ' C > (x, t) = (1 g 7/2) (1 / " /2+1) 9" «(x, t ). (5.20) ( I -? ) C'q = qcqkq, we have C'q [E, (sk~) u] (x, t) = (1 g 7/(2l - g / 7/2+1) where The expansion \ c -0 > 9,1 / (5.2D / ^r/3- j. \ [,( * jr - ) u ] ( x,t ) = J. % q)l ; * ) - (5-22) \ c y» q->1 / gives rise to the following identity: OO [E9 w] (x, t) = ^ 2 Cnhn {x, t), (5.23) n=0. ~/3. it; q~ pxt a-q t. r ' rh i c - q\~ q, 1 (a;g) m ( aqn ^ jl / 9 n,9/3 9n+7 p V = t a r o s : 1011 o v ^ t n =0 (5.24)

14 178 VIVEK SAHAI AND SHALINI SRIVASTAVA Recurrence relations. The recurrence relations that come from (5.17) are given by 20i (A; 0; x) - ga+720i (A; (3;qx) - xqx+1 13^ _ Q Q 20i (A; (3 + 1; qx) = (1 -< 2 A) 2<M A + 1;/?;x), (5.25) 20i (A;/3;qx) - qx2(j>i (A; /5; a?) = (l - qx) 20i (A - 1; /?; qx), (5.26) 6. C onclusion ( q~x q^ a+7 /3 \ ^7 ;</, q--[_q x J. In this section, we give an application of our models to g-laguerre polynomials. Let a > 1 and 0 < q < 1. Then g-laguerre polynomials are defined by [3] r a ( \ _ + )n, f q _ n+a+ A. s 9/ (qmq) ^ \ y ' 6.1) If we take 2: = (1 q) x then it can be easily seen that lim ^ i- (z; q) = (x), where (x) are classical Laguerre polynomials. Consider the model (4.15) in the particular case 7 = 1 + a (that is, a model of representation ] q ( (1 + a) / 2)) with basis functions involving g-laguerre polynomials, given as 4 = ( T(I+ ) - TzTt - ztztt), -1 _ 1 - g(l+»)/2tt / (*,*) = L (z;q)tn. (6.2) The action of these Jg-operators on fn (z, t) will take the following form: 7n+l j f f n = r (1+a)\ * q f n + 1, J f ^ r u a Jn J n 1> 1 - q 14-a 1 /Tf 2 1^ = -..f n - (6.3) We can obtain identities involving ^-Laguerre polynomials using the method explained above. For example, taking (* () = J r i A 101 ( ; 9 ~29"+a+1) <n (6 '4) n=0

15 ^-REPRESENTATIONS OF sl(2,c) 179 and proceeding as in Section 4.2.1, we arrive at the following identity: 00 fn = Ln (^<1) 1^ n= 0 ',q)n (6.5) which is a special case of (4.23). The recurrence relations involving g-laguerre polynomials can also be obtained by using first two equations of (6.3). These will shape as L (z;q )-q 1+ +nlz(qz;q)-zqn+a+1l (qz;q) = (l -,"+>) L«+1 (z; 9) (6.6) and qnlz(z;q)-lz{qz;q) = - (l - qa+n) L ^ (qz; q) (6.7) respectively. Eliminating (z; q) from (6.6) and (6.7) gives rise to a three term recurrence relation as follows (l - qa+n) _! (qz:q) + (zq1+a+2" - (l - <j1+ +2" ) ) (qz;q) + qn ( l - q n+1) K + l (z-,q) = 0, (6.8) n = 1,2,.... Note that (6.8) is a g-analogue of the three term recurrence relation for classical Laguerre polynomials [2], (.a + n) (x) 4- (x 1 - a - 2n) L% (x) + (n + 1) L +1 (x) = 0. (6.9) A cknow ledgem ents. The authors thank the referee for his valuable suggestions which led to a better presentation of the paper. R eferences 1. M.A. Al-Bassam and H.L. Manocha, Lie theory of solutions of certain differintegral operators, SIAM J. Math. Anal. 18 (1987), G.E. Andrews, R.Askey and R. Roy, Special Functions, Cambridge Univ. Press, G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge Univ. Press, M. Jimbo, A q-difference analogue of U (g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), H.L. Manocha, Lie algebras of difference differential operators and Appell function F i, J. Math. Anal. Appl. 138 (1989), H.L. Manocha, On models of irreducible q-representations of sl(2,c), Appl. Anal. 37 (1990), H.L. Manocha, and V. Sahai, On models of irreducible unitary presentations of compact Lie qroup SU(2) involvinq special functions, Indian J. Pure Appl. Math. 21 (1990), W. Miller, Lie Theory and Special Functions, Academic Press, E.D. Rainville, Special Functions, Macmillan, 1960.

16 180 VIVEK SAHAI AND SHALINI SRIVASTAVA 10. V. Sahai, Lie theory of polynomials of the form Fp via Mellin transformation, Indian J. Pure Appl. Math. 22 (1991), V. Sahai, q-representations of the Lie algebras Q{a,b), J. Indian Math. Soc. 63 (1997), V. Sahai, On models of Uq(sl(2)) and q-euler integral transformation, Indian J. Pure Appl. Math. 29 (1998), V. Sahai, On models of Uq[sl(2)) and m -fold q-euler integral transformation, J. Indian Math. Soc. 66 (2000), V. Sahai, On models of /q(sl(2)) and q-appell functions using a q-integral transformation, Proc. Amer. Math. Soc. 127 (1999), V. Sahai, On models of irreducible q-representations of sl(2, C ) by means of fractional q-calculus, New Zealand J. Math., 28(1999), H.M. Srivastava and H.L. Manocha, A treatise on generating functions, Ellis Horwood, N. Ja Vilenkin and A.U. Klimyk, Representation of Lie Groups and Special Functions, Vol. 3, Kluwer, Dordretch, L. Weisner, Group theoretic origin of certain generating functions, Pacific J. Math. 5 (1955), Vivek Sahai Department of Mathematics and Astronomy Lucknow University Lucknow INDIA vsahai@hotmail.com Shalini Srivastava Department of Mathematics and Astronomy Lucknow University Lucknow INDIA