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

Jean-Philippe Anker 1 , Fatma Ayadi 2 , and Mohamed Sifi 3
  • 1 Université d'Orléans & CNRS, Fédération Denis Poisson (FR 2964), Laboratoire MAPMO (UMR 6628), B.P. 6759, 45067 Orléans cedex 2, France
  • 2 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
  • 3 Département de Mathématiques, Université de Tunis El Manar, Faculté des Sciences de Tunis, 2092 Tunis El Manar, Tunisia

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(,).

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