21772 articles – 15587 references  [version française]
 HAL: hal-00476400, version 2
 arXiv: 1004.5203
 Adv. Pure Appl. Math. (2011) 27 pp.
 Available versions: v1 (2010-04-29) v2 (2011-01-06) v3 (2011-05-18)
 Opdam's hypergeometric functions: product formula and convolution structure in dimension 1
 (2011)
 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 $G_{\lambda}^{(\alpha,\beta)}$.
 1: Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO) Université d'Orléans – CNRS : UMR7349 2: Analyse Mathématique et Applications Ecole Préparatoire aux Etudes d'Ingénieurs de Tunis – Université Tunis El Manar
 Subject : Mathematics/Classical Analysis and ODEsMathematics/Functional Analysis
 Keyword(s): Dunkl-Cherednik operator – Opdam-Cherednik transform – product formula – convolution product – Kunze-Stein phenomenon
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 hal-00476400, version 2 http://hal.archives-ouvertes.fr/hal-00476400 oai:hal.archives-ouvertes.fr:hal-00476400 From: Fatma Ayadi <> Submitted on: Wednesday, 5 January 2011 21:40:12 Updated on: Tuesday, 19 April 2011 11:11:55