Diverse characterizations of Q-Dunkl classical orthogonal polynomials

Abstract

In this paper, we present four characterizations of q-Dunkl-classical orthogonal polynomials. The first one is a $T_{\theta ,q}$-distributional equation of Pearson type fulfilled by its associated form, where $T_{\theta ,q}$ is the q-Dunkl operator. The second is a second order linear $T_{\theta ,q}$-difference equation. The third is a first order linear $T_{\theta ,q}$-difference equation with polynomial coefficients satisfied by the corresponding Stieltjes function. The fourth is the so-called structure relation fulfilled by q-Dunkl-classical polynomials. Moreover, we provide an example of a non-symmetric sequence of $T_{\theta ,q}$-classical orthogonal polynomials.

Keywords:

• Orthogonal polynomials
• q-Dunkl operator
• regular forms
• classical polynomials

Mathematics Subject Classifications:

• Primary: 33C45
• Secondary: 42C05

Acknowledgments

The authors thank the valuable comments and suggestions by the referees. They have contributed to improve the presentation of this manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).