Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator

TL;DR

This paper introduces Clifford-Gegenbauer polynomials associated with Dunkl operators on the unit ball and Euclidean space, establishing their properties, relations to Jacobi polynomials, and orthogonality frameworks.

Contribution

It develops the theory of Clifford-Gegenbauer polynomials in the Dunkl setting, including Rodrigues formulas, differential equations, and their connection to Jacobi polynomials, expanding the understanding of Dunkl monogenics.

Findings

Derived Rodrigues formulas and differential equations for the polynomials

Established explicit relations with Jacobi polynomials

Developed a new orthogonality framework using bilinear forms

Abstract

We introduce the so-called Clifford-Gegenbauer polynomials in the framework of Dunkl operators, as well on the unit ball B(1), as on the Euclidean space $R^m$. In both cases we obtain several properties of these polynomials, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the Jacobi polynomials on the real line. As in the classical Clifford case, the orthogonality of the polynomials on $R^m$ must be treated in a completely different way than the orthogonality of their counterparts on B(1). In case of $R^m$, it must be expressed in terms of a bilinear form instead of an integral. Furthermore, in this paper the theory of Dunkl monogenics is further developed.