{
  "id": "2011.02762",
  "version": "v1",
  "published": "2020-11-05T11:21:02.000Z",
  "updated": "2020-11-05T11:21:02.000Z",
  "title": "Proof of a supercongruence conjecture of (F.3) of Swisher using the WZ-method",
  "authors": [
    "Arijit Jana",
    "Gautam Kalita"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For a non-negative integer $m$, let $S(m)$ denote the sum given by $$S(m):=\\sum_{n=0}^{m}\\frac{(-1)^n(8n+1)}{n!^3}\\left(\\frac{1}{4}\\right)_n^3.$$ Using the powerful WZ-method, for a prime $p\\equiv 3$ $($mod $4)$ and an odd integer $r>1$, we here deduce a supercongruence relation for $S\\left(\\frac{p^r-3}{4}\\right)$ in terms of values of $p$-adic gamma function. As a consequence, we prove one of the supercongruence conjectures of (F.3) posed by Swisher. This is the first attempt to prove supercongruences for a sum truncated at $\\frac{p^r-(d-1)}{d}$ when $p^r\\equiv -1$ $($mod $d)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-05T11:21:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11Y55",
      "11A07",
      "11B65",
      "40G99"
    ],
    "keywords": [
      "supercongruence conjecture",
      "adic gamma function",
      "first attempt",
      "odd integer",
      "supercongruence relation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}