{
  "id": "1912.12711",
  "version": "v1",
  "published": "2019-12-29T19:15:01.000Z",
  "updated": "2019-12-29T19:15:01.000Z",
  "title": "Positive intertwiners for Bessel functions of type B",
  "authors": [
    "Margit Rösler",
    "Michael Voit"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CA",
    "math.RT"
  ],
  "abstract": "Let $V_k$ denote Dunkl's intertwining operator for the root sytem $B_n$ with multiplicity $k=(k_1,k_2)$ with $k_1\\geq 0, k_2>0$. It was recently shown that the positivity of the operator $V_{k^\\prime\\!,k} =V_{k^\\prime}\\circ V_k^{-1}$ which intertwines the Dunkl operators associated with $k$ and $k^\\prime=(k_1+h,k_2)$ implies that $h\\in[k_2(n-1),\\infty[\\,\\cup\\,(\\{0,k_2,\\ldots,k_2(n-1)\\}-\\mathbb Z_+)$. This is also a necessary condition for the existence of positive Sonine formulas between the associated Bessel functions. In this paper we present two partial converse positive results: For $k_1 \\geq 0, \\,k_2\\in\\{1/2,1,2\\}$ and $h>k_2(n-1)$, the operator $V_{k^\\prime\\!,k}$ is positive when restricted to functions which are invariant under the Weyl group, and there is an associated positive Sonine formula for the Bessel functions of type $B_n$. Moreover, the same positivity results hold for arbitrary $k_1\\geq 0, k_2>0$ and $h\\in k_2\\cdot \\mathbb Z_+.$ The proof is based on a formula of Baker and Forrester on connection coefficients between multivariate Laguerre polynomials and an approximation of Bessel functions by Laguerre polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-12-29T19:15:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C67",
      "33C52",
      "43A85"
    ],
    "keywords": [
      "bessel functions",
      "positive intertwiners",
      "positive sonine formula",
      "positivity results hold",
      "multivariate laguerre polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}