Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator (Q550430)

Dunkl operators are combinations of differential and difference operators associated to a finite reflection group \(G\). They allow the construction of a Dunkl Laplacian which is a combination of the classical Laplacian in \(\mathbb{R}^m\) with some difference terms, such that the resulting operator is invariant under \(G\) only, not under the whole orthogonal group. Similarly to the usual Clifford analysis, the authors introduce in the framework of Dunkl operators the analogues of Clifford-Gegenbauer polynomials, both for the unit ball \(B(1)\) and for \(\mathbb{R}^m\). In both cases several properties of both polynomials are obtained. The theory of Dunkl monogenics is further developed.