{
  "id": "2102.04111",
  "version": "v1",
  "published": "2021-02-08T10:27:21.000Z",
  "updated": "2021-02-08T10:27:21.000Z",
  "title": "The Newton Polyhedron and positivity of ${}_2F_3$ hypergeometric functions",
  "authors": [
    "Yong-Kum Cho",
    "Seok-Young Chung"
  ],
  "comment": "The paper is accepted to <Constructive Approximation>",
  "journal": "Constructive Approximation, 2021",
  "categories": [
    "math.CA"
  ],
  "abstract": "As for the ${}_2F_3$ hypergeometric function of the form \\begin{equation*} {}_2F_3\\left[\\begin{array}{c} a_1, a_2\\\\ b_1, b_2, b_3\\end{array}\\biggr| -x^2\\right]\\qquad(x>0), \\end{equation*} where all of parameters are assumed to be positive, we give sufficient conditions on $(b_1, b_2, b_3)$ for its positivity in terms of Newton polyhedra with vertices consisting of permutations of $\\,(a_2, a_1+1/2, 2a_1)\\,$ or $\\,(a_1, a_2+1/2, 2a_2).$ As an application, we obtain an extensive validity region of $(\\alpha, \\lambda, \\mu)$ for the inequality \\begin{equation*} \\int_0^x (x-t)^{\\lambda}\\, t^{\\mu} J_\\alpha(t)\\, dt \\ge 0\\qquad(x>0). \\end{equation*}",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-08T10:27:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26D15",
      "33C10",
      "33C20"
    ],
    "keywords": [
      "hypergeometric function",
      "newton polyhedron",
      "positivity",
      "sufficient conditions",
      "extensive validity region"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}