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arXiv · 1003.1512

Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator

H. De Bie·N. De Schepper
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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.

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BibTeXRIS

H. De Bie, N. De Schepper. 2010-03-07. Clifford-Gegenbauer polynomials related to the Dunkl Dirac operator. https://arxiv.org/abs/1003.1512

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