{
  "id": "1004.5203",
  "version": "v3",
  "published": "2010-04-29T06:51:14.000Z",
  "updated": "2011-05-18T11:46:43.000Z",
  "title": "Opdam's hypergeometric functions: product formula and convolution structure in dimension 1",
  "authors": [
    "Jean-Philippe Anker",
    "Fatma Ayadi",
    "Mohamed Sifi"
  ],
  "comment": "Adv. Pure Appl. Math. (2011) 27 pp",
  "categories": [
    "math.CA",
    "math.FA"
  ],
  "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. R\\\"osler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\\lambda}^{(\\alpha,\\beta)}$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-05-18T11:46:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "opdams hypergeometric functions",
      "product formula",
      "dunkl-cherednik operator",
      "explicit kernel",
      "rational limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.5203A"
    }
  }
}