{
  "id": "1611.07686",
  "version": "v1",
  "published": "2016-11-23T08:35:21.000Z",
  "updated": "2016-11-23T08:35:21.000Z",
  "title": "Generalized Rodriguez-Villegas supercongruences involving $p$-adic Gamma functions",
  "authors": [
    "Ji-Cai Liu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $p\\ge 5$ be a prime and $\\langle a\\rangle_p$ denote the least non-negative integer $r$ with $a\\equiv r \\pmod{p}$. We mainly prove that for $\\langle a\\rangle_p\\equiv 0 \\pmod{2}$ \\begin{align*} \\sum_{k=0}^{p-1}\\frac{\\left(\\frac{1}{2}\\right)_k(-a)_k(a+1)_k}{(1)_k^3}\\equiv (-1)^{\\frac{p+1}{2}}\\Gamma_p\\left(-\\frac{a}{2}\\right)^2\\Gamma_p\\left(\\frac{a+1}{2}\\right)^2 \\pmod{p^2}, \\end{align*} where $(x)_k=x(x+1)\\cdots (x+k-1)$ and $\\Gamma_p(\\cdot)$ denotes the $p$-adic Gamma function. This partially extends four Rodriguez-Villegas supercongruences for truncated hypergeometric series ${}_3F_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-23T08:35:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "33C20"
    ],
    "keywords": [
      "adic gamma function",
      "generalized rodriguez-villegas supercongruences",
      "truncated hypergeometric series",
      "partially extends",
      "non-negative integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}