A NEW CHARACTERIZATION OF SYMMETRIC DUNKL AND q -DUNKL-CLASSICAL ORTHOGONAL POLYNOMIALS

: In this paper, we consider the following L -diﬀerence equation Φ( x ) L P n +1 ( x ) = ( ξ n x + ϑ n ) P n +1 ( x ) + λ n P n ( x ) , n ≥ 0 , where Φ is a monic polynomial (even), deg Φ ≤ 2, ξ n , ϑ n , λ n , n ≥ 0, are complex numbers and L is either the Dunkl operator T µ or the the q -Dunkl operator T ( θ,q ) . We show that if L = T µ , then the only symmetric orthogonal polynomials satisfying the previous equation are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials and if L = T ( θ,q ) , then the q 2 -analogue of generalized Hermite and the q 2 -analogue of generalized Gegenbauer polynomials are, up a dilation, the only orthogonal polynomials sequences satisfying the L -diﬀerence equation.


Introduction
The classical orthogonal polynomials (Hermite, Laguerre, Bessel, and Jacobi) have a lot of useful characterizations: they satisfy a Hahn's property, that the sequence of their monic derivatives is again orthogonal (see [1,8,14,16]), they are characterized as the polynomial eigenfunctions of a second order homogeneous linear differential (or difference) hypergeometric operator with polynomial coefficients [4,15,16], their corresponding linear functionals satisfy a distribution equation of Pearson type (see [11,13,15]).
Another characterization was established by Al-Salam and Chihara in [1], in particular they showed that the sequences Hermite, Laguerre and Jacobi are the only monic orthogonal polynomial sequences {P n } n≥0 that satisfy an equation of the form: where π is a monic polynomial, deg π ≤ 2.
In particular they showed that the only orthogonal polynomials satisfying (1.2) are the Al-Salam-Carlitz I, the little and big q-Laguerre, the little and big q-Jacobi and the q-Bessel polynomials.The aim of this paper is to study the equation of the form: where Φ is a monic polynomial (even), deg Φ ≤ 2 and L ∈ {T µ , T (θ,q) }.This paper is organized as follows.In Section 2, we introduce the basic background and some preliminary results that will be used in what follows.In Section 3, we show that the only symmetric orthogonal polynomials satisfying (1.3), are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials if L = T µ and the q 2 -analogue of generalized Hermite polynomials and the q 2 -analogue of generalized Gegenbauer polynomials if L = T (θ,q) .

Preliminaries and notations
Let P be the vector space of polynomials with coefficients in C and let P ′ be its dual.We denote by u, f the action of u ∈ P ′ on f ∈ P. In particular, we denote by (u) n = u, x n , n ≥ 0, the moments of u.For any form u, any polynomial f and any a ∈ C \ {0}, let f u and h a u, be the forms defined by duality: where h a p(x) = p(ax).
Let {P n } n≥0 be a sequence of monic polynomials (MPS, in short) with deg P n = n, n ≥ 0. The dual sequence associated with {P n } n≥0 is the sequence {u n } n≥0 , u n ∈ P ′ such that u n , P m = δ n,m , n, m ≥ 0, where δ n,m is the Kronecker symbol [14].
The linear functional u is called regular if there exists a MPS {P n } n≥0 such that (see [8, p. 7]): Then the sequence {P n } n≥0 is said to be orthogonal with respect to u.In this case, we have Moreover, u = λu 0 , where (u) 0 = λ = 0 [17].
In what follows all regular linear functionals u will be taken normalized i.e., (u) 0 = 1.Therefore, u = u 0 .
A polynomial set {P n } n≥0 is called symmetric if According to Favard's theorem [8], a sequence of monic orthogonal polynomials {P n (x)} n≥0 (MOPS, in short) satisfies a three-term recurrence relation with A dilatation preserves the property of orthogonality.Indeed, the sequence { P n (x)} n≥0 defined by satisfies the recurrence relation [16] P 0 (x) = 1, where Moreover, if {P n } n≥0 is a MOPS with respect to the regular form u 0 , then { P n } n≥0 is a MOPS with respect to the regular form u 0 = h a −1 u 0 .
Theorem 1 [8].Let {P n } n≥0 be a MOPS satisfying (2.1) and orthogonal with respect to a linear functional u.The following statements are equivalent: Next, we introduce the Dunkl operator T µ defined on P by [10,18] where For the Dunkl operator, we have the property [6] T In particular, We define the operator T µ from P ′ to P ′ as follows: In particular, with the convention (u) −1 = 0, where We introduce also the q-Dunkl operator T (θ,q) defined on P by [2,5,7] ( Remark 1.Note that when q → 1, we again meet the Dunkl operator.
From the last definition, it is easy to prove that In particular, We define the operator T (θ,q) from P ′ to P ′ as follows: In particular, (T (θ,q) where (u) −1 = 0 and here [n] q , n ≥ 0, denotes the basic q-number defined by According to the definitions of T µ and T (θ,q) , we have

The main results
In this section, we will look for all symmetric MOPS satisfying (1.3).We distinguish two cases.The first case is when L = T µ and the second one is when L = T (θ,q) .

First case: when
where Φ is a monic polynomial (even), deg Φ ≤ 2, are, up a dilation, the generalized Hermite polynomials and the generalized Gegenbauer polynomials.
P r o o f.Let {P n } n≥0 be a symmetric MOPS satisfying (3.1).Since Φ is a monic, even and deg Φ ≤ 2, then we distinguish two cases: Φ(x) = 1 and Φ(x) = x 2 + c.Case 1. Φ(x) = 1, then (3.1) becomes By comparing the degrees in the last equation (in x n+2 and x n+1 ), we obtain ξ n = ϑ n = 0, n ≥ 0 and then Identifying coefficients in the monomials of degree n in the last equation, we obtain On the other hand, applying the operator T µ to (2.1) with β n+1 = 0 and using (2.3), we get Substituting (3.2) and (3.3) in the last equation and taking into account the fact that From (2.1), the last equation is equivalent to Therefore, Since the last relation remains valid for n = 0, then we have Using (2.2), where a 2 = 2γ 1 /µ 1 , we obtain So, we meet the recurrence coefficients for the generalized Hermite polynomial sequence (see [8]).
Therefore, (3.6) becomes Taking into account (2.1), we get Then, Since µ n+2 − µ n = 2, then, substitution of (3.8) in (3.9) gives Therefore, So, It is clear that (3.10) remains valid for n = 0.Then, we have Substitution of (3.11) in (3.8) gives By virtue of fourth equality in (3.7), we obtain that the previous equation remains valid for n = 0. Hence, We will distinguish two situations: c = 0 and c = 0.

Second case: when
Theorem 3. The only symmetric MOPS satisfying a T (θ,q) -difference equation of the form: where Φ is a monic polynomial (even), deg Φ ≤ 2, are, up a dilation, the q 2 -analogue of generalized Hermite polynomials and the q 2 -analogue of generalized Gegenbauer polynomials.
Multiplying the above equation by x 2 + c and substituting (3.19) into the result, we get Substituting of (2.1) in the previous equation, we get The comparison of the coefficients of x n+2 in the previous equation gives ϑ n+1 = qϑ n , n ≥ 1 and putting n = 0 and n = 1 in (3.19), we get respectively Hence, ϑ n = 0, n ≥ 0. Therefore, the last equation becomes Using the fact that P n+1 = xP n (x) − γ n P n−1 , the above equation is equivalent to Then, we deduce then the substitution of (3.21) in (3.22) gives We can easily deduce by induction that It is clear that the previous identity remains valid for n = 0.Then, we have Now, we will determine λ n .By (3.23), we have Therefore, (3.21) becomes By virtue of (3.20), we obtain that the previous equation remains valid for n = 0.Then, We will distinguish two situations: c = 0 and c = 0.
Remark 3. Notice that when q → 1, we recover the result in Theorem 2 and when θ = 0 in (3.14), we again meet (1.2) for symmetric case.

Polynomial
Table 2: Case when L = T (θ,q) Remark 4. In this paper, we have studied only the symmetric case.The question for nonsymmetric case remains open.

Table 1 :
Case when L = T µ