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

Jean-Philippe Anker; Fatma Ayadi; Mohamed Sifi

doi:10.1515/apam.2011.008

Article

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

Jean-Philippe Anker , Fatma Ayadi and Mohamed Sifi

Published/Copyright: January 19, 2012

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Advances in Pure and Applied Mathematics

From the journal Volume 3 Issue 1

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-January

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Articles in the same Issue

https://doi.org/10.1515/apam.2011.008

Keywords for this article

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