1997/11/03 by Margit Rösler, Rösler, Margit, Michael Voit +1
Computer Science · Mathematics · #33C80 (Primary) 33C50 #60B15 (Secondary) #FOS: Mathematics #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.q-alg/9711004
openalex publication_date 1997/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Dunkl operators are differential-difference operators on \b RN 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 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.