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Dunkl Operators: Theory and Applications

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Orthogonal Polynomials and Special Functions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1817))

Abstract

These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunkl operators, the intertwining operator, the Dunkl kernel and the Dunkl transform. We point out the connection with integrable particle systems of Calogero-Moser-Sutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunkl kernel, the Dunkl-type heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunkl kernel.

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Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, D-37073, Göttingen, Germany

    Margit Rösler

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  1. Margit Rösler
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Editors and Affiliations

  1. Afd.Toegepaste Wiskundige Analyse, EWI-TWA,Technische Universiteit Delft, Postbus 5031, 2600, GA Delft, Netherlands

    Erik Koelink

  2. Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium

    Walter Van Assche

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Rösler, M. (2003). Dunkl Operators: Theory and Applications. In: Koelink, E., Van Assche, W. (eds) Orthogonal Polynomials and Special Functions. Lecture Notes in Mathematics, vol 1817. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44945-0_3

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  • DOI: https://doi.org/10.1007/3-540-44945-0_3

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  • Print ISBN: 978-3-540-40375-3

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