{
  "id": "1910.10932",
  "version": "v1",
  "published": "2019-10-24T06:30:46.000Z",
  "updated": "2019-10-24T06:30:46.000Z",
  "title": "A common $q$-analogue of two supercongruences",
  "authors": [
    "Victor J. W. Guo",
    "Wadim Zudilin"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We give a $q$-congruence whose specializations $q=-1$ and $q=1$ correspond to supercongruences (B.2) and (H.2) on Van Hamme's 1997 list: $$ \\sum_{k=0}^{(p-1)/2}(-1)^k(4k+1)A_k\\equiv p(-1)^{(p-1)/2}\\pmod{p^3} \\quad\\text{and}\\quad \\sum_{k=0}^{(p-1)/2}A_k\\equiv a(p)\\pmod{p^2}, $$ where $p>2$ is prime, $$ A_k=\\prod_{j=0}^{k-1}\\biggl(\\frac{1/2+j}{1+j}\\biggr)^3=\\frac1{2^{6k}}{\\binom{2k}k}^3 \\quad\\text{for}\\ k=0,1,2,\\dots, $$ and $a(p)$ is the $p$-th coefficient of (the weight 3 modular form) $q\\prod_{j=1}^\\infty(1-q^{4j})^6$. We complement our result with a general common $q$-congruence for related hypergeometric sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-24T06:30:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D15",
      "11A07",
      "11B65"
    ],
    "keywords": [
      "supercongruences",
      "van hammes",
      "th coefficient",
      "modular form",
      "general common"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}