{
  "id": "2204.04709",
  "version": "v1",
  "published": "2022-04-10T15:42:13.000Z",
  "updated": "2022-04-10T15:42:13.000Z",
  "title": "Recurrence relations of coefficients involving hypergeometric function with an application",
  "authors": [
    "Zhen-Hang Yang"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "For $a,b,p\\in \\mathbb{R}$, $-c\\notin \\mathbb{N\\cup }\\left\\{ 0\\right\\} $ and $ \\theta \\in \\left[ -1,1\\right] $, let \\begin{equation*} U_{\\theta }\\left( x\\right) =\\left( 1-\\theta x\\right) ^{p}F\\left( a,b;c;x\\right) =\\sum_{n=0}^{\\infty }u_{n}\\left( \\theta \\right) x^{n}. \\end{equation*}% In this paper, we prove that the coefficients $u_{n}\\left( \\theta \\right) $ for $n\\geq 0$ satisfies a 3-order recurrence relation. In particular, $ u_{n}\\left( 1\\right) $ satisfies a 2-order recurrence relation. These offer a new way to study for hypergeometric function. As an example, we present the necessary and sufficient conditions such that a hypergeometric mean value is Schur m-power convex or concave on $\\mathbb{R}_{+}^{2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-10T15:42:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05",
      "26B25",
      "26E60"
    ],
    "keywords": [
      "recurrence relation",
      "hypergeometric function",
      "coefficients",
      "application",
      "hypergeometric mean value"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}