Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 6 Issue 4 (2014), Pages 16-43 GENERALIZED q-HERMITE POLYNOMIALS AND THE q-DUNKL HEAT EQUATION (COMMUNICATED BY FRANCISCO MARCELLAN ) M. SALEH JAZMATI & KAMEL MEZLINI & NEJI´ BETTAIBI Abstract. Two classes of generalized discrete q-Hermite polynomials are con- structed. Several properties of these polynomials, and an explicit relations con- necting them with little q-Laguerre and q-Laguerre polynomials are obtained. A relationship with the q-Dunkl heat polynomials and the q-Dunkl associated functions are established. 1. Introduction The classical sequence of Hermite polynomials form one of the best known sys- tems of orthogonal polynomials in literature. Their applications cover many do- mains in applied and pure mathematics. For instance, the Hermite polynomials play central role in the study of the polynomial solutions of the classical heat equa- tion, which is a partial differential equation involving classical derivatives. It is therefore natural that generalizations of the heat equation for generalized oper- ators lead to generalizations of the Hermite polynomials. For instance, Fitouhi introduced and studied in [6] the generalized Hermite polynomials associated with a Sturm-Liouville operator by studying the corresponding heat equation. In [16], R¨oslerstudied the generalized Hermite polynomials and the heat equation for the Dunkl operator in several variables. In [15], Rosenblum associated Chihara's gener- alized Hermite polynomials with the Dunkl operator in one variable and used them to study the Bose-like oscillator. During the mid 1970's, G. E. Andrews started a period of very fruitful collaboration with R. Askey (see [1, 2]). Thanks to these two mathematicians, basic hypergeomet- ric series is an active field of research today. Since Askey's primary area of interest is orthogonal polynomials, q-series suddenly provided him and his co-workers, who include W. A. Al-Salam, M. E. H. Ismail, T. H. Koornwinder, W. G. Morris, D. Stanton, and J. A. Wilson, with a very rich environment for deriving q-extensions of the classical orthogonal polynomials of Jacobi, Gegenbauer, Legendre, Laguerre 2000 Mathematics Subject Classification. 33D45, 33D50, 33D90 . Key words and phrases. Generalized q-derivative operators. Generalized discrete q-Hermite polynomials. q-Dunkl heat equation. c 2014. Universiteti i Prishtin¨es,Prishtin¨e,Kosov¨e. Submitted September 23, 2014. 16 GENERALIZED q-HERMITE POLYNOMIALS AND THE q-DUNKL HEAT EQUATION 17 and Hermite. It is in this context that this paper is built around the construction of a new generalization of the Hermite polynomials, using new q-operators. In this paper, we introduce some generalized q-derivative operators with param- eter α, which, with the q-Dunkl intertwining operator (see [3]) allow us to introduce and study two families of q-discrete orthogonal polynomials that generalize the two classes of discrete q-Hermite polynomials given in [11]. Next, we consider a generalized q-Dunkl heat equation and we show that it is re- lated to the generalized discrete q-Hermite polynomials in the same way as the classical ones in [15] and in [16]. This paper is organized as follows: in Section 2, we recall some notations and useful results. In Section 3, we introduce generalized q-shifted factorials and we use them to construct some generalized q-exponential functions. In Section 4, we introduce a new class of q-derivative operators. By these operators and the q-Dunkl intertwining operator two classes of discrete q-Hermite polynomials are introduced and analyzed in Section 5. In Section 6, we introduce a q-Dunkl heat equation, and we construct two basic sets of solutions of the q-Dunkl heat equation: the set of generalized q-heat polynomials and the set of generalized q-associated functions. In particular, we show that these classes of solutions are closely related to the generalized discrete q-Hermite I polynomials and the generalized discrete q-Hermite II polynomials, respectively. 2. Notations and Preliminaries 2.1. Basic symbols. We refer to the general reference [9] for the definitions, notations and properties of the q-shifted factorials and the basic hypergeometric series. n Throughout this paper, we fix q 2 (0; 1) an we write Rq = {±q ; n 2 Zg. For a complex number a, the q-shifted factorials are defined by: n−1 1 Y k Y k (a; q)0 = 1; (a; q)n = (1 − aq ); n = 1; 2; :::;(a; q)1 = (1 − aq ): k=0 k=0 (a; q)1 (a; q)n = n : (2.1) (aq ; q)1 If we change q by q−1, we obtain −1 −1 n − n(n−1) (a; q )n = (a ; q)n(−a) q 2 ; a 6= 0: (2.2) We also denote 1 − qx (q; q) [x] = ; x 2 and n! = n ; n 2 : q 1 − q C q (1 − q)n N The basic hypergeometric series is defined by k(k−1) 1+s−r 1 k (a1; q)k:::(ar; q)k (−1) q 2 X k rφs (a1; :::; ar; b1; :::; bs; q; z) = z : (b1; q)k:::(bs; q)k(q; q)k k=0 18 M. SALEH JAZMATI & KAMEL MEZLINI & NEJI´ BETTAIBI The two Euler's q-analogues of the exponential function are given by ( see [9]) 1 k(k−1) X q 2 k Eq(z) = z = (−z; q)1; (2.3) (q; q)k k=0 1 X 1 k 1 eq(z) = z = ; jzj < 1: (2.4) (q; q)k (z; q)1 k=0 Note that the function Eq(z) is entire on C. But for the convergence of the sec- ond series, we need jzj < 1; however, because of its product representation, eq is −n continuable to a meromorphic function on C with simple poles at z = q ; n non-negative integer. We denote by 1 X zn exp (z) := e ((1 − q)z) = ; (2.5) q q n! n=0 q 1 n(n−1) n X q 2 z Exp (z) := E ((1 − q)z) = : (2.6) q q n! n=0 q z z We have lim expq(z) = lim Expq(z) = e , where e is the classical exponential q!1− q!1− function. The Rubin's q-exponential function is defined by (see [13]) 1 2 X 2 e(z; q ) = bn(z; q ); (2.7) n=0 where [ n ]([ n ]+1) 2 q 2 2 n bn(z; q ) = z (2.8) n!q and [x] is the integer part of x 2 R. 2 2 z e(z; q ) is entire on C and we have lim e(z; q ) = e . q!1− The q-Gamma function is given by (see [5, 9] ) (q; q)1 1−z Γq(z) = z (1 − q) ; z 6= 0; −1; −2;::: (2.9) (q ; q)1 and tends to Γ(z) when q tends to 1−. We shall need the Jackson q-integrals defined by (see [9, 10]) Z a 1 Z b Z b Z a X n n f(x)dqx = (1 − q)a f(aq )q ; f(x)dqx = f(x)dqx − f(x)dqx; 0 n=0 a 0 0 Z 1 1 X n n f(x)dqx = (1 − q) f(q )q ; (2.10) 0 n=−∞ Z 1 1 1 X n n X n n f(x)dqx = (1 − q) q f(q ) + (1 − q) q f(−q ): −∞ n=−∞ n=−∞ 1 We denote by Lα,q the space of complex-valued functions f on Rq such that Z 1 2α+1 jf(x)jjxj dqx < 1: −∞ GENERALIZED q-HERMITE POLYNOMIALS AND THE q-DUNKL HEAT EQUATION 19 The q-Gamma function has the q-integral representation (see [5, 8]) Z (1−q)−1 z−1 Γq(z) = x Expq(−qx)dqx; (2.11) 0 and satisfies the relation Z 1 2 2 2 2x+1 (q t ; q )1 Γq2 (x + 1)Γq2 (y) t 2y 2 2 dqt = ; x > −1; y > 0: (2.12) 0 (q t ; q )1 (1 + q)Γq2 (x + y + 1) 2.2. The q-derivatives. The Jackson's q-derivative Dq (see [9, 10]) is defined by : f(z) − f(qz) D f(z) = : (2.13) q (1 − q)z + We also need a variant Dq , called forward q-derivative, given by f(q−1z) − f(z) D+f(z) = : (2.14) q (1 − q)z + 0 Note that lim Dqf(z) = lim Dq f(z) = f (z) whenever f is differentiable at z. q!1− q!1− Recently, R. L. Rubin introduced in [13, 14] a q-derivative operator @q as follows + @qf = Dq fe + Dqfo; (2.15) where fe and fo are respectively the even and the odd parts of f: − 0 We note that if f is differentiable at x then @qf(x) tends as q ! 1 to f (x). 3. The generalized q-exponential functions 3.1. The generalized q-shifted factorials. For α > −1, we define the generalized q-shifted factorials for non-negative integers n by 2n 2 2 2α+2 2 (1 + q) Γq2 (α + n + 1)Γq2 (n + 1) (q ; q )n(q ; q )n (2n)!q,α := = 2n ; Γq2 (α + 1) (1 − q) 2n+1 2 2 2α+2 2 (1 + q) Γq2 (α + n + 2)Γq2 (n + 1) (q ; q )n(q ; q )n+1 (2n + 1)!q,α := = 2n+1 : Γq2 (α + 1) (1 − q) We denote 2 2 2α+2 2 2 2 2α+2 2 (q; q)2n,α = (q ; q )n(q ; q )n and (q; q)2n+1,α = (q ; q )n(q ; q )n+1: Remarks. 1 (1) If α = − , then we get 2 (q; q) 1 = (q; q)n and n! 1 = n!q: (3.1) n;− 2 q;− 2 (2) We have the recursion relations (n + 1)!q,α = [n + 1 + θn(2α + 1)]q n!q,α and (3.2) (q; q)n+1,α = (1 − q)[n + 1 + θn(2α + 1)]q (q; q)n,α; 20 M.