Communications in Mathematical Physics

, Volume 192, Issue 3, pp 519–542

Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators

  • Margit Rösler

DOI: 10.1007/s002200050307

Cite this article as:
Rösler, M. Comm Math Phys (1998) 192: 519. doi:10.1007/s002200050307


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 ℝN. 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 the case of the symmetric group SN, 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.

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Margit Rösler
    • 1
  1. 1.Zentrum Mathematik, Technische Universität München, Arcisstr. 21, D-80333 München, Germany.¶E-mail: roesler@mathematik.tu-muenchen.deDE