A new characterization of symmetric dunkl and $q$-dunkl-classical orthogonal polynomials

Yahia Habbachi

Ural mathematical journal, Tome 9 (2023) no. 2, pp. 109-120

Résumé

In this paper, we consider the following $\mathcal{L}$-difference equation $$\Phi(x) \mathcal{L}P_{n+1}(x)=(\xi_nx+\vartheta_n)P_{n+1}(x)+\lambda_nP_{n}(x),\quad n\geq0,$$ where $\Phi$ is a monic polynomial (even), $\deg\Phi\leq2$, $\xi_n,\,\vartheta_n,\,\lambda_n,\,n\geq0$, are complex numbers and $\mathcal{L}$ is either the Dunkl operator $T_\mu$ or the the $q$-Dunkl operator $T_{(\theta,q)}$. We show that if $\mathcal{L}=T_\mu$, 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 $\mathcal{L}=T_{(\theta,q)}$, 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 $\mathcal{L}$-difference equation.

Détail

Keywords: Dunkl operator, $q$-Dunkl operator.
Mots-clés : Orthogonal polynomials