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Biorthogonal polynomials associated with reflection groups and a formula of Macdonald

✍ Scribed by Margit Rösler; Michael Voit

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
714 KB
Volume
99
Category
Article
ISSN
0377-0427

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✦ Synopsis

Dunkl operators are differential-difference 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 polynomials with respect to Dunkl's Gaussian distributions in a 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. (~

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