{
  "id": "2204.01045",
  "version": "v1",
  "published": "2022-04-03T10:56:18.000Z",
  "updated": "2022-04-03T10:56:18.000Z",
  "title": "When does a hypergeometric function ${}_{p\\!}F_q$ belong to the Laguerre--Pólya class $LP^+$?",
  "authors": [
    "Alan D. Sokal"
  ],
  "comment": "10 pages, LaTeX2e",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "I show that a hypergeometric function ${}_{p}F_q(a_1,\\ldots,a_p;b_1,\\ldots,b_q;\\,\\cdot\\,)$ with $p \\le q$ belongs to the Laguerre--P\\'olya class $LP^+$ for arbitrarily large $b_{p+1},\\ldots,b_q > 0$ if and only if, after a possible reordering, the differences $a_i - b_i$ are nonnegative integers. This result arises as an easy corollary of the case $p=q$ proven two decades ago by Ki and Kim. I also give explicit examples for the case ${}_{1}F_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-03T10:56:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C20",
      "30B70",
      "30C15",
      "30D20",
      "30E05",
      "44A60"
    ],
    "keywords": [
      "hypergeometric function",
      "laguerre-pólya class",
      "laguerre-polya class",
      "result arises",
      "explicit examples"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}