21829 articles – 15616 references  [version française]

hal-00476400, version 2

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

Jean-Philippe Anker 1, Fatma Ayadi () 1, Mohamed Sifi 2

Adv. Pure Appl. Math. (2011) 27 pp.

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 $G_{\lambda}^{(\alpha,\beta)}$.

  • 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
  • Domain : Mathematics/Classical Analysis and ODEs
    Mathematics/Functional Analysis
  • Keywords : Dunkl-Cherednik operator – Opdam-Cherednik transform – product formula – convolution product – Kunze-Stein phenomenon
  • Available versions :  v1 (2010-04-29) v2 (2011-01-06) v3 (2011-05-18)
 
  • 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