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

Abstract : Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an integral in terms of $G_{\lambda}^{(\alpha,\beta)}(z)$ 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 $G_{\lambda}^{(\alpha,\beta)}$.
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https://hal.archives-ouvertes.fr/hal-00476400
Contributor : Jean-Philippe Anker <>
Submitted on : Wednesday, May 18, 2011 - 1:42:13 PM
Last modification on : Thursday, June 6, 2019 - 12:08:02 PM
Document(s) archivé(s) le : Friday, August 19, 2011 - 2:23:26 AM

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Jean-Philippe Anker, Fatma Ayadi, Mohamed Sifi. Opdam's hypergeometric functions: product formula and convolution structure in dimension 1. Advances in Pure and Applied Mathematics, De Gruyter, 2012, 3 (1), pp.11-44. ⟨10.1515/apam.2011.008⟩. ⟨hal-00476400v3⟩

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