Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 093, 11 pages      arXiv:0812.4819      https://doi.org/10.3842/SIGMA.2008.093
Contribution to the Special Issue on Dunkl Operators and Related Topics

An Alternative Definition of the Hermite Polynomials Related to the Dunkl Laplacian

Hendrik De Bie
Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Krijgslaan 281, 9000 Gent, Belgium

Received October 07, 2008, in final form December 18, 2008; Published online December 28, 2008

Abstract
We introduce the so-called Clifford-Hermite polynomials in the framework of Dunkl operators, based on the theory of Clifford analysis. Several properties of these polynomials are obtained, such as a Rodrigues formula, a differential equation and an explicit relation connecting them with the generalized Laguerre polynomials. A link is established with the generalized Hermite polynomials related to the Dunkl operators (see [Rösler M., Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.]) as well as with the basis of the weighted L2 space introduced by Dunkl.

Key words: Hermite polynomials; Dunkl operators; Clifford analysis.

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