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

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 [R\"osler M., Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006]) as well as with the basis of the weighted $L^{2}$ space introduced by Dunkl.Comment: This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA