GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION (COMMUNICATED BY FRANCISCO MARCELLAN )

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1 Bulletin of Mathematical Analysis and Applications ISSN: , URL: Volume 6 Issue 4 14), Pages GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION COMMUNICATED BY FRANCISCO MARCELLAN ) M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI Abstract Two classes of generalized discrete q-hermite polynomials are constructed Several properties of these polynomials, and an explicit relations connecting 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 systems of orthogonal polynomials in literature Their applications cover many domains in applied and pure mathematics For instance, the Hermite polynomials play central role in the study of the polynomial solutions of the classical heat equation, which is a partial differential equation involving classical derivatives It is therefore natural that generalizations of the heat equation for generalized operators 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], Rösler studied the generalized Hermite polynomials and the heat equation for the Dunkl operator in several variables In [15], Rosenblum associated Chihara s generalized Hermite polynomials with the Dunkl operator in one variable and used them to study the Bose-like oscillator During the mid 197 s, G E Andrews started a period of very fruitful collaboration with R Askey see [1, ]) Thanks to these two mathematicians, basic hypergeometric 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 Mathematics Subject Classification 33D45, 33D5, 33D9 Key words and phrases Generalized q-derivative operators Generalized discrete q-hermite polynomials q-dunkl heat equation c 14 Universiteti i Prishtinës, Prishtinë, Kosovë Submitted September 3, 14 16

2 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 parameter 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 related 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, 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 Notations and Preliminaries 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 Throughout this paper, we fix q, 1) an we write R q = {±q n, n Z} For a complex number a, the q-shifted factorials are defined by: n 1 a; q) = 1; a; q) n = 1 aq k ), n = 1,, ; a; q) = 1 aq k ) If we change q by q 1, we obtain We also denote a; q) n = a; q) aq n ; q) 1) a; q 1 ) n = a 1 ; q) n a) n q nn 1), a ) [x] q = 1 qx 1 q, x C and n! q = q; q) n 1 q) n, n N The basic hypergeometric series is defined by ) 1+s r a 1 ; q) k a r ; q) k 1) k q kk 1) rφ s a 1,, a r ; b 1,, b s ; q, z) = z k b 1 ; q) k b s ; q) k q; q) k

3 18 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI The two Euler s q-analogues of the exponential function are given by see [9]) q kk 1) E q z) = z k = z; q), 3) q; q) k e q z) = 1 z k 1 =, z < 1 4) q; q) k z; q) Note that the function E q z) is entire on C But for the convergence of the second series, we need z < 1; however, because of its product representation, e q is continuable to a meromorphic function on C with simple poles at z = q n, n non-negative integer We denote by z n exp q z) := e q 1 q)z) =, 5) n! q Exp q z) := E q 1 q)z) = q nn 1) z n n! q 6) We have lim exp qz) = lim Exp qz) = e z, where e z is the classical exponential q 1 q 1 function The Rubin s q-exponential function is defined by see [13]) ez; q ) = b n z; q ), 7) where b n z; q ) = q[ n ][ n ]+1) n! q z n 8) and [x] is the integer part of x R ez; q ) is entire on C and we have lim q 1 ez; q ) = e z The q-gamma function is given by see [5, 9] ) Γ q z) = q; q) q z ; q) 1 q) 1 z, z, 1,, 9) and tends to Γz) when q tends to 1 We shall need the Jackson q-integrals defined by see [9, 1]) a b b fx)d q x = 1 q)a faq n )q n, fx)d q x = fx)d q x fx)d q x = 1 q) a fx)d q x = 1 q) n= n= q n fq n ) + 1 q) a fx)d q x, fq n )q n, 1) n= q n f q n ) We denote by L 1 α,q the space of complex-valued functions f on R q such that fx) x α+1 d q x <

4 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 19 The q-gamma function has the q-integral representation see [5, 8]) and satisfies the relation Γ q z) = q) 1 t x+1 q t ; q ) q y t ; q ) d q t = The q-derivatives The Jackson s q-derivative D q see [9, 1]) is defined by : D q fz) = x z 1 Exp q qx)d q x, 11) Γ q x + 1)Γ q y), x > 1, y > 1) 1 + q)γ q x + y + 1) fz) fqz) 13) 1 q)z We also need a variant D + q, called forward q-derivative, given by D q + fz) = fq 1 z) fz) 14) 1 q)z Note that lim D q fz) = lim D + q 1 q 1 q fz) = f z) whenever f is differentiable at z Recently, R L Rubin introduced in [13, 14] a q-derivative operator q as follows q f = D + q f e + D q f o, 15) where f e and f o are respectively the even and the odd parts of f We note that if f is differentiable at x then q fx) tends as q 1 to f x) 3 The generalized q-exponential functions 31 The generalized q-shifted factorials For α > 1, we define the generalized q-shifted factorials for non-negative integers n by n)! q,α := 1 + q)n Γ q α + n + 1)Γ q n + 1) Γ q α + 1) n + 1)! q,α := 1 + q)n+1 Γ q α + n + )Γ q n + 1) Γ q α + 1) We denote = q ; q ) n q α+ ; q ) n 1 q) n, = q ; q ) n q α+ ; q ) n+1 1 q) n+1 q; q) n,α = q ; q ) n q α+ ; q ) n and q; q) n+1,α = q ; q ) n q α+ ; q ) n+1 Remarks 1) If α = 1, then we get ) We have the recursion relations q; q) n, 1 = q; q) n and n! q, 1 = n! q 31) and n + 1)! q,α = [n θ n α + 1)] q n! q,α q; q) n+1,α = 1 q) [n θ n α + 1)] q q; q) n,α, 3)

5 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI where 1 if n is even θ n = if n is odd 3) It is easy to prove the following limits 33) lim n)! q,α = n n!γα + n + 1) = γ q 1 Γα + 1) α+ 1 n), 34) lim q 1 n + 1)! q,α = n+1 n!γα + n + ) = γ Γα + 1) α+ 1 n + 1), where γ µ is the Rosenblum s generalized factorial see [15]) 3 The generalized q-exponential functions By means of the generalized q-shifted factorials, we construct three generalized q-exponential functions as follows: Definition 31 For z C, the generalized q-exponential functions are defined by where E q,α z) := e q,α z) := ψ α,q λ q kk 1) z k q; q) k,α 35) z k q; q) k,α, z < 1 36) z) = b n,α iλz; q ), λ C, 37) b n,α z; q ) = q[ n ][ n ]+1) z n n! q,α 38) Note that ψ α,q λ z) is exactly the q-dunkl kernel introduced in [3] For α = 1, it follows from 31) that E q, 1 z) = E qz), e q, 1 z) = e qz), b n, 1 x; q ) = b n x; q ) and we have ψ 1,q λ x) = eiλx; q ) the Rubin s q-exponential function 7) By 34), the q-dunkl kernel ψ α,q λ x) tends to e α+ 1 iλx) as q 1, where e µ is the Rosenblum s generalized exponential function see [15, 16]) The generalized q-exponential function ψ α,q λ x) gives rise to a q-integral transform, called the q-dunkl transform on the real line, which was introduced and studied in [3]: where F α,q D f)λ) = K α + fx)ψ α,q λ x) x α+1 d q x, K α = 1 q)α q α+ ; q ) q ; q ) f L 1 α,q,

6 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 1 4 The generalized q-derivatives In this section we introduce a new class of q-derivatives operators which play an important role in the construction of a generalized q-hermite polynomials 41 The generalized q-derivative operators Definition 41 The generalized backward and forward q-derivative operators D q,α and D + q,α are defined as The operators given by D q,α fz) = fz) qα+1 fqz), 41) 1 q)z D q,αfz) + = fq 1 z) q α+1 fz) 4) 1 q)z are called the generalized q-derivatives operators α,q f = D q f e + D q,α f o, 43) + α,qf = D + q f e + D + q,αf o 44) Remark that for α = 1, we have: D q, 1 = D q, D + = D q, 1 q +, q, 1 = D q and + = D q, 1 q + The following elementary result is useful in the sequel Lemma 41 + α,qx n = q n [n + θ n+1 α + 1)] q x n 1, n = 1,, 3, 45) + α,q,xe q,α qxt) = t 1 q e q,αtx), 46) + [ α,q,x Eq q x )e q,α qxt) ] = t x 1 q E q q x )e q,α tx), 47) where the operator + α,q,x acts with respect to x and θ is given by 33) Proof By elementary calculus, we get 45) On the one hand, using the definition of the operator + α,q and 45), we have + α,q,xe q,α qxt) = qt) n + α,qx n q; q) n,α = n=1 t n [n + θ n+1 α + 1)] q x n 1 q; q) n,α On the other hand, using the second recursion relation in 3) and changing the index in the second sum in the previous equation, we obtain + α,q,xe q,α qxt) = t t n x n = t 1 q q; q) n,α 1 q e q,αxt) Finally, using the infinite product representation 3) of E q q x ), we get + [ α,q,x Eq q x )e q,α qxt) ] [ = E q q x ) + α,qe q,α qxt) x ] 1 q e q,αxt), and 47) follows from 46)

7 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI 4 The q-dunkl operator We can rewrite the q-dunkl operator introduced in [3] by means of the generalized q-derivative operators introduced in Definition 41 as Indeed We have and where We can write, then, Λ α,q f = + α,qf e + α,q f o 48) q f = D + q f e + D q f o, fx) f x) Λ α,q f)x) = q [H α,q f)] x) + [α + 1] q, x H α,q : f = f e + f o f e + q α+1 f o Λ α,q fx) = q f e x) + q α+1 q f o x) + 1 qα+1 1 q)x f ox) = D + q f e x) + q α+1 D q f o x) + 1 qα+1 1 q)x f ox) = D + q f e x) + D q,α f o x) = + α,qf e x) + α,q f o x) It is noteworthy that in the case α = 1, Λ α,q reduces to the Rubin s q-derivative operator q defined in [13] and that for a differentiable function f, the q-dunkl operator Λ α,q f tends to the classical Dunkl operator Λ α f as q tends to 1 By induction, we prove the following results : Proposition 41 1) A repeated application of the q-dunkl operator to the monomial b n,α x; q ) gives Λ k α,qb n,α x; q ) = b n k,α x; q ), k =, 1, n 49) ) If f is an even function, then Λ n α,qfx) = q nn+1) n α,qfq n x), n =, 1,,, 41) Λ n+1 α,q fx) = q n+1) n+1 α,q fq n 1 x); n =, 1,, 411) 43 The q-dunkl intertwining operator For our further development, we need to extend the notion of the q-dunkl intertwining operator introduced in [3] to the space CR) of continuous functions on R Definition 4 The q-dunkl intertwining operator V α,q is defined on CR) by where and V α,q fx) = Cα; q) 1 W α t; q )1 + t)fxt)d q t, x R, 41) Cα; q) = 1 + q)γ qα + 1) Γ q 1/)Γ q α + 1/) W α t; q ) = 413) t q ; q ) t q α+1 ; q ) 414)

8 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 3 Since the integrand is continuous, the q-integral 41) is well-defined In the following proposition we shall show that V α,q is the intertwining operator between the generalized q-derivatives and the usual q-derivatives which generalize the transmutation relation 53) in [3] Proposition 4 Suppose that the function f and its q-derivatives D q f, D + q f and q f are in CR), then α,q V α,q f) = V α,q D q f); 415) + α,qv α,q f) = V α,q D + q f); 416) Λ α,q V α,q f) = V α,q q f) 417) Proof By splitting f into its even and odd parts f = f e + f o, and using the fact that D q changes the parity of the function and the q-integral of an odd function on [ 1, 1] is equal zero, we obtain V α,q D q f)x) = Cα; q) Cα; q) + 1 W α t; q )td q f e xt)d q t 1 W α t; q )D q f xt)d q t On the other hand, from the definition of α,q 43) we have α,q V α,q f)x) = Cα; q) Cα; q) + 1 W α t; q )td q f e xt)d q t 1 W α t; q )t D q,α f xt)d q t Then, using the fact that 1 q α+1 t )W α t; q ) = 1 q t )W α qt; q ), we get V α,q D q f)x) α,q V α,q f)x) = = = ) 419) W α t; q ) [ D q f xt) t D q,α f xt) ] d q t W α t; q ) 1 t )f xt) d q t 1 q)xt W α t; q ) 1 qα+1 t )f qxt) d q t 1 q)xt W α t; q ) 1 t )f xt) q)xt d q t W α qt; q ) 1 q t )f qxt) d q t 1 q)xt Since the integrand in the last Jackson q-integral vanishes at the points 1 and 1, the change of variable u = qt in this q-integral leads to the relation 415) In a similar way, one can obtain 416) Now by using 48), 415) and 416), we obtain Λ α,q V α,q f) = + α,qv α,q f e + α,q V α,q f o = V α,q D + q f e + D q f o ) = Vα,q q f) The following result shows the effect of the q-dunkl intertwining operator on monomial functions and on the q-exponential functions

9 4 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI Proposition 43 The following relations hold: V α,q z n = q; q) n q; q) n,α z n, n =, 1,, ; 4) V α,q b n z; q ) = b n,α z; q ), n =, 1,, ; 41) V α,q e q z) = e q,α z), z < 1; 4) V α,q E q z) = E q,α z); 43) V α,q eλz; q ) = ψ α,q iλ z), λ C 44) Proof If we take x = n 1 and y = α + 1 in 1), we get by using 413) V α,q z n = Cα; q)z n t q ; q ) t q α+1 ; q t n d q t = Cα; q) Γ qn + 1 )Γ qα + 1 ) ) 1 + q)γ q n + α + 1) zn = Γ q α + 1)Γ q n + 1 ) Γ q 1 )Γ q n + α + = q; q ) n 1)zn q α+ ; q z n ) n Similarly, we prove that V α,q z n+1 = Cα; q)z n+1 Then 4 ) follows from the two following facts q; q) n = q; q ) n q; q) n,α q α+ ; q, ) n t q ; q ) t q α+1 ; q ) t n+1) d q t = q; q ) n+1 q α+ ; q ) n+1 z n+1 q; q) n+1 q; q) n+1,α = q; q ) n+1 q α+ ; q ) n+1 45) For z < 1, we have by using 4) V α,q z k z k V α,q e q z) = = = e q,α z) q; q) k q; q) k,α The same techniques produce the relations 43) and 44) 5 The generalized discrete q-hermite polynomials We begin this section by the following useful lemma Lemma 51 1) For s, we have F q s) := t s E q q t )d q t = 1 q) q ; q ) q s+1 ; q ) 51) ) For λ > and n non-negative integer, we have G α q,nλ) := where e q λy )y n+α+1 d q y = c q,α λ) q n α+1)n λ n q α+ ; q ) n, 5) c q,α λ) = 1 q) qα+ λ, q α /λ, q ; q ) λ, q /λ, q α+ ; q ) 53)

10 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 5 Proof 1) Let s From 11), we have by doing the two changes of variables u = 1 q )t and u = t, Γ q s) = 1 1 q ) s t s 1 E q q t)d q t = 1 + q 1 q ) s u s 1 E q q u )d q u The relation 51) follows then by replacing s by s + 1 in the previous equation ) Let λ > and n non negative integer Then, from the definition of the Jackson q-integral and the Ramanujan identity see [9], p 15), we have e q λy )y s q s+1)k d q y = 1 q) λq k ; q ) In particular for s = n + α + 1, we have k= = 1 q) λq s+1, q s+1 /λ, q ; q ) λ, q s+1, q /λ; q ) e q λy )y n+α+1 d q y = 1 q) λq α+n+, q α n /λ, q ; q ) λ, q α+n+, q /λ; q ) We conclude 5) by using the following equalities: λq α+n+ ; q ) = q α+ λ; q ) q α+ λ; q ) n, q α+n+ ; q ) = q α+ ; q ) q α+ ; q ) n and q α n /λ; q ) = q nn+1) q α λ) n q α+ λ; q ) n q α λ) 1 ; q ) 51 The generalized discrete q-hermite I polynomials The discrete q-hermite I polynomials {h n x; q)} are defined in [11] by h n x; q) := x n φ q n, q n+1 ; ; q, q n 1 x ) = q; q) n [ n ] 1) k q kk 1) x n k q ; q ) k q; q) n k 54) Definition 51 The generalized discrete q-hermite I polynomials {h n,α x; q)} are defined by: Remarks h n,α x; q) := q; q) n [ n ] 1) For α = 1, we get h n, 1 x; q) = h nx; q) ) Observe that h n,α 1 q x; q) 1 q ) n = n! q 1 + q) n [ n ] 1) k q kk 1) x n k q ; q ) k q; q) n k,α 55) 1) k q kk 1) 1 + q)x) n k k! q n k)! q,α

11 6 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI So by 34), we obtain h n,α 1 q x; q) lim q 1 1 q ) n 1 = Hα+ n x) n, where H α+ 1 n x) is the Rosenblum s generalized Hermite polynomial see [15] ) 3) Each polynomial h n,α ; q) has the same parity of its degree n Lemma 5 The generalized discrete q-hermite I polynomials can be written in terms of basic hypergeometric functions as: h n,α x; q) = q; q) n x n φ q n, q n α ; ; q, q 4n+α x ) q; q) n,α = 1) n q nn 1) q; q ) n φ 1 q n, ; q α+ ; q, q x ), h n+1,α x; q) = q; q) n+1 q; q) n+1,α x n+1 φ q n, q n α ; ; q, q 4n+α+ x ) Proof We have Using the identity we get and It follows, then, that = 1) n q nn 1) q; q ) n+1 1 q α+ x φ 1 q n, ; q α+4 ; q, q x ) h n,α x; q) = q; q) n a; q ) n k = n 1) k q kk 1) x n k q ; q ) k q ; q ) n k q α+ ; q ) n k 56) a; q ) n a 1 q n ; q ) k q a 1) k q kk 1) nk, 57) q ; q ) n k = q ; q ) n q n ; q ) k 1) k q kk 1) nk 58) q α+ ; q ) n k = qα+ ; q ) n q n α ; q ) k q α ) k q kk 1) nk 59) h n,α x; q) = q; q) n q; q) n,α x n Now, we have n 1) k q kk 1) q n ; q ) k q n α ; q ) k q 4n+α x ) k q ; q ) k = q; q) n q; q) n,α x n φ q n, q n α ; ; q, q 4n+α x ) h n+1,α x; q) = q; q) n+1 n 1) k q kk 1) x n+1 k q ; q ) k q ; q ) n k q α+ ; q ) n+1 k

12 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 7 Then, using 58) and replacing n by n + 1 in 59), we obtain and q α+ ; q ) n+1 k = qα+ ; q ) n+1 q n α ; q ) k q α ) k q kk 1) nk k h n+1,α x; q) = q; q) n+1 q; q) n+1,α x n+1 n 1) k q kk 1) q n ; q ) k q n α ; q ) k q 4n+α+ x ) k q ; q ) k = q; q) n+1 q; q) n+1,α x n+1 φ q n, q n α ; ; q, q 4n+α+ x ) On the other hand, taking b = q n α and z = q n+α x in the following transformation formula see [11], p19) φ q n, b; ; q, q n z) = b; q ) n z n φ 1 q n, ; b 1 q n ; q, q ), 51) bz we obtain φ q n, q n α ; ; q, q 4n+α x ) = q n α ; q ) n q n +αn x n φ 1 q n, ; q α+ ; q, q x ) By the identity see [11], p9) we get a; q ) n = a 1 q n ; q ) n a) n q nn 1), a, 511) φ q n, q n α ; ; q, q 4n+α x ) = q α+ ; q ) n 1) n q nn 1) x n φ 1 q n, ; q α+ ; q, q x ) So, it follows from 45) that h n,α x; q) = 1) n q nn 1) q; q ) n φ 1 q n, ; q α+ ; q, q x ) Now, take b = q n α and z = q n+α+ x in 51), to obtain φ q n, q n α ; ; q, q 4n+α+ x ) = q n α ; q ) n q n +αn+n x n φ 1 q n, ; q α+4 ; q, q x ) By identity 511), we have φ q n, q n α ; ; q, q 4n+α+ x ) = q α+4 ; q ) n 1) n q nn 1) x n and by 45), we get φ 1 q n, ; q α+4 ; q, q x ), h n+1,α x; q) = 1) n q nn 1) q; q ) n+1 1 q α+ x φ 1 q n, ; q α+4 ; q, q x ) The little q-laguerre polynomials {p n x; a q)} are defined in [11] by p n x; a q) = φ 1 q n, ; aq; q, qx) 51)

13 8 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI Using 539 ) and 51), the generalized discrete q-hermite I polynomials h n,α x; q) can be expressed in terms of the little q-laguerre polynomials p n x; a q) as follows: h n,α x; q) = 1) n q nn 1) q; q ) n p n x ; q α q ), h n+1,α x; q) = 1) n q nn 1) q; q ) n+1 1 q α+ xp nx ; q α+ q ) Proposition 51 The following relations hold: 1) q-integral representation of Mehler type ) Generating function 3) Inversion formula E q z )e q,α xz) = h n,α x; q) = V α,q h n x; q) 513) x n = q; q) n,α 4) Three terms recursion formula z n h n,α x; q), xz < 1) 514) q; q) n [ n ] h n k,α x; q) q ; q ) k q; q) n k 515) xh n,α x; q) q n 1+α+1)θn 1 q n )h n 1,α x; q) = 1 qn+1+α+1)θn 1 q n+1 h n+1,α x; q) 516) Proof 1) The relation 513) follows by application of 4 ) to each term in 54) ) The generating function for the discrete q-hermite I polynomials is given by see [11]) E q z z n )e q xz) = h n x; q) 517) q; q) n So, by applying the operator V α,q, with respect to x, to both sides of equation 517) and using 513) and 4), we obtain 514) 3) follows by application of the operator V α,q to both sides of the well-known relation see [1]): x n = q; q) n [ n ] h n k x; q) q ; q ) k q; q) n k, 4) and 513) 4) To prove 516), we consider, separately, the even and the odd cases in the expression xh n,α x; q) q n 1+α+1)θn 1 q n )h n 1,α x; q) We have xh n,α x; q) q n+α 1 q n )h n 1,α x; q) n 1) k q kk 1) x n k+1 = q; q) n q ; q ) k q; q) n k,α n 1 q n+α 1) k q kk 1) x n k+1)+1 q; q) n q ; q ) k q; q) n k+1)+1,α

14 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 9 Change k to k 1 in the second sum, then combine with the first to get q; q) n x n+1 n 1) k q kk 1) x n k+1 + q; q) n q; q) n,α q ; q ) k q; q) n k+1,α k=1 [ 1 q n k+α+ ) + q n+α q k+ 1 q k ) ] Simplify to obtain q; q) n x n+1 q; q) n,α + 1 q n+α+ )q; q) n The last expression can be written as n k=1 1) k q kk 1) x n k+1 q ; q ) k q; q) n k+1,α 1 q n+α+ 1 q n+1 h n+1,α x; q) In the odd case, the proof follows the same steps as the even case The following result states the affect of the operators α,q and + α,q on the generalized q-hermite I polynomials Proposition 5 1) The forward shift operator: or equivalently α,q h n,α x; q) = 1 qn 1 q h n 1,αx; q), n = 1,, 518) h n,α x; q) q α+1)θn+1 h n,α qx; q) = 1 q n )xh n 1,α x; q) 519) ) The backward shift operator: + [ α,q Eq q x )h n,α x; q) ] = q n [θ n α + 1) + n + 1] q 1 q n+1 E q q x )h n+1,α x; q), or equivalently 5) q θn+1α+1) h n,α x; q) 1 x )h n,α q 1 x; q) = q n [θ n α + 1) + n + 1] q [n + 1] q xh n+1,α x; q) Proof 1) It is well known see [11]) that D q h n x; q) = 1 qn 1 q h n 1x; q) So, by application of the q-dunkl intertwining operator to both sides of this result, we obtain 518), by the use of 415) and 513) ) Put gx, t) = E q q x )E q q t )e q,α qxt) From 47), we have + α,q,xgx, t) = E q q t ) t x 1 q E q q x )e q,α xt) = E q q x ) x t 1 q E q q t )e q,α xt) = + α,q,tgx, t) 51)

15 3 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI But, using the generating function 517), we obtain + α,q,xgx, t) = + [ α,q,x Eq q x )h n,α x; q) ] qt) n 5) q; q) n and + α,q,xgx, t) = E q q x q n )h n,α x; q) + q; q) α,qt n n It follows from 45 ) and 51) that + E q q x )h n,α x; q) α,q,xgx, t) = [θ n+1 α + 1) + n] q t n 1 53) q; q) n n=1 Therefore, by comparing the coefficients of t n in the two series 5) and 53), we obtain 5) The following formula can now be proved by induction Proposition 53 The Rodrigues-formula for {h n,α x; q)} is given by E q q x )h n,α x; q) = q 1)n q nn 1) q; q) n q; q) n,α [ + α,q ] n [ Eq q x ) ] 54) Proof Since h,α x; q) = 1, the formula is clearly true for n = Assume that it is true for an integer n Then, using 5) and 3), the application of the operator + α,q to the both sides of 54) completes the induction proof Proposition 54 The polynomials {h n,α x; q)} satisfy the following q-difference equations: q α+1 h n,α qx; q) q + q α+1 q 1 n x )h n,α x; q) + q1 x )h n,α q 1 x; q) = 55) q α+1 h n+1,α qx; q) 1+q α+ q n x )h n+1,α x; q)+q1 x )h n+1,α q 1 x; q) = 56) Proof By 518), we have α,qh n,α x; q) = 1 qn )1 q n 1 ) 1 q) h n,α x; q), n = 1,, 3, But, from the three recursion term formulas 516), we obtain xh n 1,α x; q) q n 1 q n 1 )h n,α x; q) = h n,α x; q) 57) and from 519), we get h n,α x; q) h n,α qx; q) = 1 q n )xh n 1,α x; q) 58) It follows, then, from 57) and 58) that or equivalently, α,qh n,α x; q) = 1 [ q 1 q) h n,α x; q) q n h n,α qx; q) ] 1 q x )h n,α x; q) 1+q α q n x )h n,α qx; q)+q α h n,α q x; q) = 59)

16 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 31 Replace x by q 1 x in 59) and multiply the result by q, we obtain 55) Following the same steps, we prove 56) To prove the orthogonality property of h n,α x; q), we need the following lemma: Lemma 53 The polynomials {h n,α x; q)} satisfy x p h n,α x; q)e q q x ) x α+1 d q x 1 1 q)q ; q ) = q α+ ; q q; q) n q [ n ][ n+1 ]+α) if p = n, ) if p =, 1,, n 1 53) Proof Since the parity of the q-polynomials {h n,α x; q)} is the parity of thier degrees and the q-integral in 53) of odd function is zero It is, then, sufficient to consider the cases where p and n are both even or odd From the definition 54), we can write 1 x p h n,α x; q)e q q x ) x α+1 d q x = q; q) n n 1) n k q n k)n k 1) F q k + p + α + 1) q ; q ) n k q ; q ) k q α+ ; q ) k, where F q is the function defined by 51), then the above sum becomes 1 q)q ; q ) q; q) n n 531) 1) n k q n k)n k 1) q ; q ) n k q ; q ) k q α+ ; q ) k q k+p+α+ ; q ) Using 1), it is possible to rewrite the sum in 531) in the form 1 q)q ; q ) q; q) n n 1) n k q n k)n k 1) q α+ ; q ) q ; q ) n k q ; q q k+α+ ; q ) p ) k By using the transformation formulas 58) and the fact that the sum in 531) can be written as q k+α+ ; q ) p = qα+ ; q ) p q p+α+ ; q ) k q α+ ; q ) k, 53) 1) n q nn 1) 1 q)q ; q ) q; q) n q α+ ; q ) p q α+ ; q ) q ; q ) n n q n ; q ) k q p+α+ ; q ) k q ; q ) k q α+ ; q q k ) k = 1)n q nn 1) 1 q)q ; q ) q; q) n q α+ ; q ) p q α+ ; q ) q ; q ) n φ 1 q n, q p+α+ ; q α+ ; q, q ) By the summation formula see [11], p15) φ 1 q n, b; c; q, q ) = b 1 c; q ) n c; q ) n b n,

17 3 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI the sum in 531) is equal to 1) n q nn 1) 1 q)q ; q ) q; q) n q α+ ; q ) p q α+ ; q ) q ; q ) n q p ; q ) n q α+ ; q ) n q p+α+)n So, if p < n, the q-integral in 531) vanishes If p = n, and the q-integral in 531) is equal to q n ; q ) n = q ; q ) n 1) n q n nn 1), 533) 1 q)q ; q ) q; q) n q n +αn q α+ ; q ) In the case where both p and n are odd integers the proof follows the same steps as in the first case 1 x p+1 h n+1,α x; q)e q q x ) x α+1 d q x = q; q) n+1 n 1) n k q n k)n k 1) F q k + p + α + 3) q ; q ) n k q ; q ) k q α+ ; q ) k+1 = 1 q)q ; q ) q; q) n+1 n 1) n k q n k)n k 1) q ; q ) n k q ; q ) k q α+ ; q ) k+1 q k+p+α+4 ; q ) = 1)n q nn 1) 1 q)q ; q ) q; q) n+1 q α+4 ; q ) p q α+ ; q ) q ; q ) n φ 1 q n, q p+α+4 ; q α+4 ; q, q ) = 1)n q nn 1) 1 q)q ; q ) q; q) n+1 q α+4 ; q ) p q α+ ; q ) q ; q ) n q p ; q ) n q α+4 ; q ) n q p+α+4)n So, if p < n, the above q-integral vanishes If p = n, by 533) the q-integral is equal to 1 q)q ; q ) q; q) n+1 q n +αn+n q α+ ; q ) Theorem 51 For n, m =, 1,,, we have the orthogonality relation 1 h m,α x; q)h n,α x; q)e q q x ) x α+1 d q x = 1 q)q ; q ) q; q) nq [ n ][ n+1 ]+α) q α+ ; q δ n,m ) q; q) n,α 534)

18 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 33 Proof Without loss of generality, we assume that m n By 55), the q-integral in 534) is equal to q; q) m [ m ] p= 1) p q pp 1) q ; q ) p q; q) m p,α Then, Lemma 53) completes the proof of the theorem 1 x m p h n,α x; q)e q q x ) x α+1 d q x 5 The generalized discrete q-hermite II polynomials Recall that the discrete q-hermite II polynomials { h n x; q)} are defined by see [11]) h n x; q) = x n φ q n, q n+1 ; ; q, q x ) = q; q) n [ n ] 1) k q nk q kk+1) x n k q ; q ) k q; q) n k The generating function for the discrete q-hermite II polynomials is see [11]) e q z )E q xz) = 535) q nn 1) hn x; q)z n 536) q; q) n The monomial function can be expressed in terms of the discrete q-hermite II polynomials as see [11]) x n = q; q) n [ n ] q nk+3k h n k x; q) q ; q ) k q; q) n k 537) In the following definition, we introduce a new generalization of the discrete q- Hermite II polynomials Definition 5 The generalized discrete q-hermite II polynomials { h n,α x; q)} are defined by Remarks n [ ] h n,α x; q) := q; q) n 1) k q nk q kk+1) x n k q ; q ) k q; q) n k,α 538) 1) It is easy to see that for α = 1, h n, 1 x; q) = h n x; q) ) Note that h n,α 1 q x; q) 1 q ) n = n! q 1 + q) n [ n ] and by 34), we get hn,α 1 q x; q) lim q 1 1 q ) n 1) k q nk q kk+1) 1 + q)x) n k k! q n k)! q,α, 1 = Hα+ n x) n, where H µ n is the Rosenblum s generalized Hermite polynomial see [15])

19 34 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI Lemma 54 The generalized discrete q-hermite II polynomials can be written in terms of basic hypergeometric functions as: h n,α x; q) = q; q) n x n φ 1 q n, q n α ; ; q, q α+3 x ) q; q) n,α = 1) n q nn 1) q; q ) n 1 φ 1 q n ; q α+ ; q, q n+1 x ) ; h n+1,α x; q) = q; q) n+1 q; q) n+1,α x n+1 φ 1 q n, q n α ; ; q, q α+3 x ) Proof We have = 1) n q nn+1) x q; q ) n+1 1 q α+ 1 φ 1 q n ; q α+4 ; q, q n+3 x ) h n,α x; q) = q; q) n n Using the identities 58) and 59), we obtain and h n,α x; q) = q; q) n q; q) n,α x n 1) k q 4nk q kk+1) x n k q ; q ) k q ; q ) n k q α+ ; q ) n k n q n ; q ) k q n α ; q ) k q α+3 x ) k q ; q ) k = q; q) n q; q) n,α x n φ 1 q n, q n α ; ; q, q α+3 x ) h n+1,α x; q) = q; q) n+1 q; q) n+1,α x n+1 539) n q n ; q ) k q n α ; q ) k q α+3 x ) k q ; q ) k = q; q) n+1 q; q) n+1,α x n+1 φ 1 q n, q n α ; ; q, q α+3 x ) But, we have see [11], p16) ) φ 1 q n, a 1 q n ; ; q, aqn+ = a; q ) n q n z n 1φ 1 q n ; a; q, z ) 54) z So, taking a = q α+ and z = q n+1 x in 54), we get φ 1 q n, q n α ; ; q, q α+3 x ) = 1) n q n +n x n q α+ ; q ) n 1 φ 1 q n ; q α+ ; q, q n+1 x ) and taking a = q α+4 and z = q n+3 x in 54), we obtain φ 1 q n, q n α ; ; q, q α+3 x ) = 1) n q n n x n q α+4 ; q ) n which achieves the proof 541) 1 φ 1 q n ; q α+4 ; q, q n+3 x ), 54) We recall that the q-laguerre polynomials L α) n x; q) are defined by see [11]) L α) n x; q) = qα+1 ; q) n q; q) n 1φ 1 q n ; q α+1 ; q, q n+α+1 x)

20 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 35 Using 539) the generalized discrete q-hermite II polynomials h n,α x; q) can also be expressed in terms of the q-laguerre polynomials L α) n x; q) as follows: h n,α x; q) = 1) n q nn 1) q; q) n q α+ ; q L α) n q α 1 x ; q ), ) n h n+1,α x; q) = 1) n q nn+1) q; q) n+1 q α+ ; q ) n+1 xl α+1) n q α 1 x ; q ) Proposition 55 The following properties hold: 1) The generalized discrete q-hermite II polynomials h n,α x; q) are related to the generalized discrete q-hermite I polynomials h n,α x; q) by h n,α x; q) = i) n q α+ 1 )θn+1 h n,α ixq α 1 ; q 1 ) 543) ) q-integral representation of Mehler type 3) Generating function 4) Inversion formula: h n,α x; q) = V α,q hn x; q) 544) e q z )E q,α xz) = x n = q; q) n,α [ n ] 5) Three terms recursion formula q nn 1) hn,α x; q)z n 545) q; q) n q nk+3k hn k,α x; q) q ; q ) k q; q) n k 546) x h n,α x; q) q 1 n 1 q n ) h n 1,α x; q) = 1 qn+1+α+1)θn 1 q n+1 hn+1,α x; q) 547) Proof 1) 543) follows from the relation ) ) Applications of the operator V α,q to each term in 535) and the result 4) give 544) 3) By application of the operator V α,q to both sides of equation 536) and using 513) and 43), we obtain 545) 4) Using the same argument to both sides of result 537), 4) and 544) gives 546) 5) By considering separately the even and the odd cases in x h n,α x; q) q 1 n 1 q n ) h n 1,α x; q), 547) follows from 538) by elementary calculus In the following proposition we derive some of the important properties of generalized discrete q-hermite II polynomials h n,α x; q) from those of generalized discrete q-hermite I polynomials h n,α x; q) by using the identity 543)

21 36 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI Proposition 56 The generalized discrete q-hermite II polynomials h n,α x; q) satisfy the following properties: 1) Affect of the forward shift operator h n,α x; q) q α+1)θn+1 h n,α qx; q) = q n+1 1 q n )x h n 1,α qx; q) 548) or equivalently n+1 1 qn α,q hn,α x; q) = q 1 q h n 1,α qx; q), n = 1,, 549) ) Affect of the backward shift operator h n,α x; q) q α+1)θn q α 1 x ) h n,α qx; q) = q n 1 q n 1 α+1)θn 1 q n 1 x h n+1,α x; q) or equivalently ] α,q [ω α x; q) h n,α x; q) = q n 1 q n 1 α+1)θn 1 q)1 q n 1 ) ω αx; q) h n+1,α x; q), 55) where 3) q-difference equations ω α x; q) = e q q α 1 x ) 551) 1+q α 1 x ) h n,α qx; q) 1+q α +q n α 1 x ) h n,α x; q)+q α hn,α q 1 x; q) = and 1 + q α 1 x ) h n+1,α qx; q) q + q α 1 + q n α 1 x ) h n+1,α x; q) The following formula can be proved by induction: Proposition 57 Rodrigues-type formula +q α hn+1,α q 1 x; q) = ω α x; q) h n,α x; q) = q 1)n q nn 1) q 1 ; q 1 ) n q 1 ; q 1 ) n,α n α,qω α x; q), 55) where ω α x; q) is the function given by 551) Proof Since h,α x; q) = 1, the formula is clearly true for n = Assume that 55) is true for an integer n Then, applying the operator α,q to both sides of equation 55) and using 55), we get ω α x; q) h n+1,α x; q) = q n 1 q)1 q n 1 ) [ ] α,q ω α x; q) h n,α x; q) 1 q n 1 α+1)θn = q n 1 q)1 q n 1 ) q 1) n nn 1) q q 1 ; q 1 ) n 1 q n 1 α+1)θn q 1 ; q 1 ) n,α n+1 α,q ω α x; q) Finally, the use of 3), where q is replaced by q 1, shows that 55) is also true for n + 1) This achieves the proof To establish an orthogonality relation for the q-polynomials { h n,α x; q)}, we need the following lemma

22 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 37 Lemma 55 The polynomials { h n,α x; q)} satisfy x p hn,α x; q)ω α x; q) x α+1 d q x q n 1 q) q, q, q ; q ) q; q) n = q α 1, q α+3, q α+ ; q if p = n, ) if p =, 1,, n 1 553) Proof As in the proof of 53), we consider two cases following q-integral First, we compute the x p hn,α x; q)ω α x; q) x α+1 d q x 554) From the definition 538), the q-integral 554) is equal to q; q) n 1) n q n +n n 1) k q k k G α q,k+p q α 1 ) q ; q ) n k q ; q ) k q α+ ; q ) k, where G α q,k+pq α 1 ) = c q,α q α 1 )q k+p) q α+ ; q ) is given by 5) k+p Then the above sum becomes n c q,α q α 1 )q; q) n 1) n q n +n 1) k q k k q k+p) q ; q ) n k q ; q ) k q α+ ; q ) k q α+ ; q ) k+p Using 58) and the fact that q α+ ; q ) = q p+α+ ; q ) k+p k q α+ ; q ), 555) p the above sum can be written as c q,α q α 1 ) 1) n q n +n p q; q ) n q α+ ; q ) p n q n ; q ) k q p+α+ ; q ) k q ; q ) k q α+ ; q q n p)k ) k = c q,α q α 1 ) 1) n q n +n p q; q ) n q α+ ; q ) p φ 1 q n, q p+α+ ; q α+ ; q, q n p ) By the summation formula see [11], p15) the q-integral in 554) becomes φ 1 q n, b; c; q, cqn b ) = b 1 c; q ) n c; q, ) n c q,α q α 1 ) 1) n q n +n p q; q ) n q α+ ; q ) p q p ; q ) n q α+ ; q ) n The only case that the above product is not zero is when p = n, where we have c q,α q α 1 )q; q ) n 1) n q 3n +n q n ; q ) n

23 38 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI By using the fomula 533), the q-integral in 554) equals 1 q) q, q, q ; q ) q α 1, q α+3, q α+ ; q q 4n q; q) ) n In the case where both p and n are odd integers the proof follows the same steps as in the first case x p+1 hn+1,α x; q)ω α x; q) x α+1 d q x = q; q) n+1 1) n q n n n 1) k q k +k G α q,k+p+1 q α 1 ) q ; q ) n k q ; q ) k q α+ ; q ) k+1, where G α q,k+p+1q α 1 ) = c q,α q α 1 )q k+p+1) q α+ ; q ) k+p+1 5) Then the above sum becomes n c q,α q α 1 )q; q) n+1 1) n q n n 1) k q k +k q k+p+1) q ; q ) n k q ; q ) k q α+ ; q ) k+1 q α+ ; q ) k+p+1 Using 58) and 555), it can be written as 556) is given by c q,α q α 1 ) 1) n q n n p+1) q; q ) n+1 q α+ ; q ) p+1 n q n ; q ) k q p+α+4 ; q ) k q ; q ) k q α+ ; q q n p)k ) k+1 = c q,α q α 1 ) 1) n q n n p+1) q; q ) n+1 q α+4 ; q ) p n q n ; q ) k q p+α+4 ; q ) k q ; q ) k q α+4 ; q q n p)k ) k = c q,α q α 1 ) 1) n q n n p+1) q; q ) n+1 q α+4 ; q ) p φ 1 q n, q p+α+4 ; q α+4 ; q, q n p ) = c q,α q α 1 ) 1) n q n n p+1) q; q ) n+1 q α+4 ; q ) p q p ; q ) n q α+4 ; q ) n The only case that the above product is not zero is when p = n, and in this case the q-integral in 556 ) equals c q,α q α 1 )q; q ) n+1 1) n q 3n 3n 1 q n ; q ) n By using the fomula 533), the q-integral in 556 ) equals 1 q) q, q, q ; q ) q α 1, q α+3, q α+ ; q q n+1) q; q) ) n+1

24 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 39 Theorem 5 The sequence of the q-polynomials { h n,α x; q)}, satisfies the orthogonality relation h n,α x; q) h m,α x; q)ω α x; q) x α+1 d q x = q n 1 q) q, q, q ; q ) q α 1, q α+3, q α+ ; q ) q; q) n q; q) n,α δ n,m 557) Proof Without loss of generality, we assume that m n, by 538) the q-integral in 557) is equal q; q) m [ m ] p= 1) p q mp q pp+1) q ; q x m p hn,α x; q)ω α x; q) x α+1 d q x ) p q; q) m p,α Then the result 553) concludes the proof of the theorem 6 The q-dunkl heat equation Let us consider the following q-dunkl heat equation: D q,tu = Λ α,q,xu, 61) where D q,t is the partial q -derivative in time defined by 13) and Λ α,q,x is the q-dunkl operator in space defined by 48) Taking into account that lim D q q 1,tut, x) = u t, x) and lim q 1 Λ α,q,xut, x) = Λ α,xut, x), t and clearly, the standard Heat equation for Dunkl operator is recovered when q 1 see [15, 16]) If α = 1, then 61) reduces to the q-heat equation D q,tu = q,xu studied in [7] 61 The q-dunkl heat polynomials We define the q-dunkl heat polynomial of degre n as v n,α x, t; q) = n! q We easily derive from 49) that and we have [ n ] t k b n k,α x; q ) 6) k! q Λ α,x v n,α x, t; q) = [n] q v n 1,α x, t; q), 63) D q,tv n,α x, t; q) = [n] q [n 1] q v n,α x, t; q) 64) It Follows from 63) and 64) that all q-dunkl heat polynomials v n,α x, t; q) are solutions of 61) Note that when α = 1, v n, 1 x, t; q) = v nx, t; q) the q-heat polynomials defined in [7] According to the limit in 34), the q-dunkl heat polynomials are the q- analogue of the generalized heat polynomials introduced by M Rosenblum in [15]

25 4 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI Proposition 61 The generating function of the sequence {v n,α x, t; q)} is given by exp q tz ) ψx α,q iz) = v n,α x, t; q) zn 65) n! q The expansion of a monomial in terms of the v n,α x, t; q) is given by b n,α x; q ) = [ n ] t) k q kk 1) v n k,α x, t; q) k! q n k)! q 66) Proof 65) follows by using 5) and 37) and expanding the q-exponential functions in 65) as power series, and taking the Cauchy product of the results Multiply both sides of 65) with Exp q tz ) and next compare the coefficients of z n, we obtain 66) It is not very difficult to verify that the q-dunkl heat polynomials are closely related to the q-polynomials h n,α x; q) as follows: Proposition 6 v n,α x, t; q) = q nn 1) iβ) n h n,αiβq n x; q), 67) where v n+1,α x, t; q) = q n iβ) n+1 h n+1,αiβq n x; q), 68) 1 q β = βt, q) = t1 + q) 69) 6 The q-dunkl heat kernel In order to find the q-dunkl heat kernel of 61), we suggest to apply the q-dunkl transform as described in [7] for q-rubin Fourier transform to the function k α x, t; q) defined by k α x, t; q) = C α t; q)exp q q α x ) t1 + q), x R, t >, 61) where Since and C α t; q) = q α 1 q) t1+q) 1 q), qα+ t1+q) 1 q), q α+ ; q ) 611) q 1 q) t1+q), t1+q) 1 q), q ; q ) Λ α,x k α x, t; q) = D q + x k α x, t; q) = q α+ 1 + q)t k αq 1 x, t; q), 61) Λ α,xk α x, t; q) = [ ] q α 1 q α+ x k α q 1 x, t; q) 1 + q)t 1 q t1 + q) = D q,tk α x, t; q) k α x, t; q) is solution of q-dunkl heat equation 61) and called the q-dunkl heat kernel It generalizes the q-source solution of q-heat equation kx, t; q) studied in [7] by taking α = 1 in 61)

26 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 41 Proposition 63 The q-dunkl heat kernel k α x, t; q) has the following properties: k α x, t; q)b n,α x; q ) x α+1 d q x = tn, t >, n! q n =, 1, 613) F α,q D k α, t; q)x) = K α exp q tx ) 614) Proof Take λ = q α 1 q) in 5), to get t1 + q) ) e q q α 1 q) y y α+n+1 d q y = c q,α λ) q nn+1) 1 + q) n t n q α+ t1 + q) 1 q) n ; q ) n, where c q,α λ) = 1 q) q 1 q) t1+q), t1+q) 1 q), q ; q ) q α 1 q) t1+q), qα+ t1+q) 1 q), q α+ ; q ) 615) Now, by definitions of k α x, t; q) 61) and of b n,α x; q ) 38) the result 613) follows By definition of ψ α,q x y) in 37), and since k α y, t; q) is even in y, we have, by using the Fubini theorem, F α,q D k α, t; q))x) = K α k α y, t; q) b n,α ixy; q ) y α+1 d q y So, by 613), we obtain F α,q D = K α 1) n x n k α y, t; q)b n,α y; q ) y α+1 d q y The conclusion follows from 5) k α, t; q))x) = K α 1) n x n tn n! q 63 The q-associated functions for q-dunkl operator We extend the notion of the q-associated functions defined in [7] by defining the q-dunkl associated functions in terms of the q-dunkl derivatives of the q-dunkl heat kernel as follows: w n,α x, t; q) = 1 + q)) n Λ n α,qk α x, t; q), n =, 1,,, t > Consequently, the q-dunkl associated functions are solutions of q-dunkl heat equation 61) Proposition 64 For n =, 1,, Λ n α,qk α x, t; q) = qnn 1) nα+1) q; q) n,α t n 1 q ) n q; q) n hn,α q n γx; q)k α q n x, t; q), 616) α,q k α x, t; q) = qn 1 n+1)α+1) q; q) n+1,α t n+ 1 1 q ) n+ 1 q; q) n+1 h n+1,α q n 1 γx; q)k α q n 1 x, t; q), Λ n+1 617)

27 4 M SALEH JAZMATI & KAMEL MEZLINI & NÉJI BETTAIBI where γ = q1 q) t1 + q) Proof Referring to 61) and 551), we can write Using 41) and 411), we obtain and k α x, t; q) = C α t; q)ω α γx; q) Λ n α,qk α x, t; q) = C α t; q)λ n α,qω α γx; q) = C α t; q)q nn+1) γ n [ n α,qω α ]γq n x; q), Λ n+1 α,q k α x, t; q) = C α t; q)λ n+1 α,q ω α γx; q) = C α t; q)q n+1) γ n+1 [ α,q n+1 ω α ]γq n 1 x; q) From Rodrigues-type formula 55) we have and Λ n α,qk α x, t; q) = γn q nn+1) q 1) n q nn 1) q 1 ; q 1 ) n,α q 1 ; q 1 ) n h n,α q n γx; q)k α q n x, t; q), Λ n+1 α,q k α x, t; q) = γn+1 q n+1) q 1) n 1 q nn+1) q 1 ; q 1 ) n+1,α q 1 ; q 1 ) n+1 h n+1,α q n 1 γx; q)k α q n 1 x, t; q) Use the fact that q 1 ; q 1 ) n,α q 1 ; q 1 ) n = q nα+1) q; q) n,α q; q) n, to obtain the desired result q 1 ; q 1 ) n+1,α q 1 ; q 1 ) n+1 = q n+1)α+1) q; q) n,α q; q) n, The above calculation shows that the q-dunkl associated functions are closely related to the q-polynomials h n,α x; q) as follows: Proposition 65 For n =, 1,, w n,α x, t; q) = qnn α 1) q; q) n,α tγ) n q; q) n hn,α q n γx; q)k α q n x, t; q), 618) w n+1,α x, t; q) = qn+1)n α 1) q; q) n+1,α tγ) n+1 hn+1,α q n 1 γx; q)k α q n 1 x, t; q), q; q) n+1 619) where q1 q) γ = t1 + q)

28 GENERALIZED q-hermite POLYNOMIALS AND THE q-dunkl HEAT EQUATION 43 References [1] G E Andrews, q-series: their development and application in analysis, number theory, combinatorics, physics, and computer algebra, Regional Conf Ser in Math, no 66, Amer Math Soc, Providence, R I 1986 [] G E Andrews and R Askey, Enumeration of partitions: The role of Eulerian series and q-orthogonal polynomials, Higher combinatorics, M Aigner, ed, Reidel, 1977), pp 3-6 [3] N Bettaibi and R Bettaieb, q-analogue of the Dunkl Transform on the Real Line, Tamsui Oxford Journal of Mathematical Sciences 5), ) [4] L Dhaouadi, A Fitouhi, J El Kamel, Inequalities in q-fourier Analysis, J Inequ Pure Appl Math, V7, Isuue 5, Article 171 6) [5] A De Sole and V G Kac, On integral representations of a q-gamma and a q-beta functions, Atti Accad Naz Lincei CI Sci Fis Mat Natur Rend Lincei9) Mat App, 16, ) [6] A Fitouhi, Heat polynomials for singular differential operator on, + ), Constr Approx ), 41-7 [7] A Fitouhi, N Bettaibi and K Mezlini, On a q-analogue of the one-dimensional heat equation Bulletin of Mathematical Analysis and Applications 4), ) [8] A Fitouhi, M M Hamza and F Bouzeffour, The q j α Bessel function, J Approx Theory, 115, ) [9] G Gasper, M Rahmen, Basic Hypergeometric Series, Encyclopedia of Mathematics and its application Vol, 35, Cambridge Univ Press, Cambridge, UK, second ed, 4 [1] V G Kac, P Cheung, Quantum Calculs, Universitext, Springer-Verlag, New York, [11] R Koekoek and R F Swarttouw, The Askey scheme of hypergeometric orthogonal polynomials and its q-analogue, Report 98-17, Delft University of Technology, Faculty of Technical Mathematics and Informatics, 1998 [1] T H Koornwinder, Special Functions and q-commuting Variables, in Special Functions, q- Series and related Topics, M E H Ismail, D R Masson and M Rahman eds), Fields Institute Communications 14, American Mathematical Society, pp ; arxiv:q-alg/ ) [13] Richard L Rubin A q -Analogue Operator for q -analogue Fourier Analysis J Math Analys App 1, ) [14] Richard L Rubin Duhamel Solutions of non-homogenous q -Analogue Wave Equations Proc of Amer Maths Soc 1353), ) [15] M Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, In: Operator theory: Advances and Applications, Vol 73, Basel: Birkhäuser Verlag ) [16] M Rösler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Commun Math Phys 19, ) Department of Mathematics, College of Science, Qassim University, KSA address: jazmati@yahoocom Department of Mathematics, University College in Leith, Umm Al-Qura University, KSA address: kamelmezlini@lamsinrnutn N Bettaibi College of Science, Qassim university, Burida, KSA address: nejibettaibi@ipeinrnutn