Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

Abstract

Based on the theory of Dunkl operators, this paper presents a general concept of multivariable Hermite polynomials and Hermite functions which are associated with finite reflection groups on R N . The definition and properties of these generalized Hermite systems extend naturally those of their classical counterparts; partial derivatives and the usual exponential kernel are here replaced by Dunkl operators and the generalized exponential kernel K of the Dunkl transform. In case of the symmetric group S N , our setting includes the polynomial eigenfunctions of certain Calogero-Sutherland type operators. The second part of this paper is devoted to the heat equation associated with Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem is governed by a positive one-parameter semigroup; this is assured by a maximum principle for the generalized Laplacian. The explicit solution to the Cauchy problem involves again the kernel K, which is, on the way, proven to be nonnegative for real arguments.

Riesz transforms for the Dunkl harmonic oscillator

“…In the setting of general Dunkl's theory Rösler [9] constructed systems of naturally associated multivariable generalized Hermite polynomials and Hermite functions. The construction starts from an arbitrarily chosen orthonormal basis {ϕ n } of P equipped with the generalized Fischer inner product…”

Generalized Fock spaces and Weyl commutation relations for the Dunkl kernel

“…admits a unique analytic solution on R d , which will be denoted E k (x, y) and called the Dunkl kernel ( [17,18,19] and [20]). This kernel has a unique holomorphic extension to…”

“…From (2), the Dunkl kernel E k possesses the following properties ( [17,19] and [20]): For all z, w ∈ C d and λ ∈ C,…”

Bessel and Flett Potentials associated with Dunkl Operators on $\Bbb R^d$

“…where △ k is applied to x variables (see [9]). The function F k t may be called the heat kernel associated with Dunkl operators or the k-heat kernel and it has the following basic properties.…”