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

TL;DR

This paper introduces Clifford-Hermite polynomials within Dunkl operator theory, establishing their properties and connections to generalized Laguerre and Hermite polynomials, enriching the mathematical framework of Dunkl analysis.

Contribution

It defines Clifford-Hermite polynomials related to Dunkl operators and explores their fundamental properties and connections to existing polynomial families.

Findings

Derived Rodrigues formula for Clifford-Hermite polynomials

Established differential equation satisfied by these polynomials

Connected Clifford-Hermite polynomials with generalized Laguerre and Hermite polynomials

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.