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Article Dans Une Revue Advances in Pure and Applied Mathematics Année : 2012

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

Résumé

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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Dates et versions

hal-00476400 , version 1 (26-04-2010)
hal-00476400 , version 2 (05-01-2011)
hal-00476400 , version 3 (18-05-2011)

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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, 2012, 3 (1), pp.11-44. ⟨10.1515/apam.2011.008⟩. ⟨hal-00476400v3⟩
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