{
  "id": "1401.5978",
  "version": "v2",
  "published": "2014-01-23T14:06:26.000Z",
  "updated": "2014-08-03T14:46:59.000Z",
  "title": "Some q-analogues of supercongruences of Rodriguez-Villegas",
  "authors": [
    "Victor J. W. Guo",
    "Jiang Zeng"
  ],
  "comment": "14 pages, to appear in J. Number Theory",
  "doi": "10.1016/j.jnt.2014.06.002",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}^2}{16^k} \\equiv (-1)^{\\frac{p-1}{2}}\\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \\sum_{k=0}^{p-1}\\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\\varepsilon)k} &\\equiv (-1)^{\\frac{p-1}{2}}q^{\\frac{(p^2-1)\\varepsilon}{4}}\\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \\varepsilon=\\pm1. Several related conjectures are also proposed.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-03T14:46:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A10",
      "05A30"
    ],
    "keywords": [
      "rodriguez-villegas",
      "supercongruences",
      "odd prime"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.5978G"
    }
  }
}