Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator

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 $\mathbb{R}^m$. 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 $\mathbb{R}^m$ must be treated in a completely different way than the orthogonality of their counterparts on $B(1)$. In case of $\mathbb{R}^m$, 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.