Biorthogonal Polynomials Associated with Reeection Groups and a Formula of Macdonald

Abstract

Dunkl operators are diierential-diierence operators on R N which generalize partial derivatives. They lead to generalizations of Laplace operators, Fourier transforms, heat semigroups, Hermite polynomials, and so on. In this paper we introduce two systems of biorthogonal poly-nomials with respect to Dunkl's Gaussian distributions in a quite canonical way. These systems, called Appell systems, admit many properties known from classical Hermite polynomials, and turn out to be useful for the analysis of Dunkl's Gaussian distributions. In particular, these polynomials lead to a new proof of a generalized formula of Macdonald due to Dunkl. The ideas for this paper are taken from recent works on non-Gaussian white noise analysis and from the umbral calculus.

Cite this paper

@inproceedings{Osler1997BiorthogonalPA, title={Biorthogonal Polynomials Associated with Reeection Groups and a Formula of Macdonald}, author={Margit Rr Osler and Michael Voit}, year={1997} }