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Advances in Pure and Applied Mathematics

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Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

Jean-Philippe Anker
  • Université d'Orléans & CNRS, Fédération Denis Poisson (FR 2964), Laboratoire MAPMO (UMR 6628), B.P. 6759, 45067 Orléans cedex 2, France
  • :
/ Fatma Ayadi
  • Département de Mathématiques, Université de Tunis El Manar, 2092 Tunis El Manar, Tunisia; and Université d'Orléans & CNRS, Fédération Denis Poisson (FR 2964), Laboratoire MAPMO (UMR 6628), B.P. 6759, 45067 Orléans cedex 2, France
  • :
/ Mohamed Sifi
  • Département de Mathématiques, Université de Tunis El Manar, Faculté des Sciences de Tunis, 2092 Tunis El Manar, Tunisia
  • :
Published Online: 2012-01-19 | DOI: https://doi.org/10.1515/apam.2011.008

Abstract.

Let G(,) be the eigenfunctions of the Dunkl–Cherednik operator T(,) on . In this paper we express the product G(,)(x)G(,)(y) as an integral in terms of G(,)(z) with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. Rösler for the Dunkl kernel. We then define and study a convolution structure associated to G(,).

Keywords.: Dunkl–Cherednik operator; Opdam–Cherednik transform; product formula; convolution product; Kunze–Stein phenomenon


Received: 2010-04-30

Revised: 2011-04-12

Published Online: 2012-01-19

Published in Print: 2012-01-01


Citation Information: Advances in Pure and Applied Mathematics. Volume 3, Issue 1, Pages 11–44, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152, DOI: https://doi.org/10.1515/apam.2011.008, January 2012

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