{
  "id": "1806.02735",
  "version": "v1",
  "published": "2018-06-07T15:43:07.000Z",
  "updated": "2018-06-07T15:43:07.000Z",
  "title": "A supercongruence concerning truncated hypergeometric series ${}_nF_{n-1}$",
  "authors": [
    "Chen Wang",
    "Hao Pan"
  ],
  "comment": "This is a preliminary manuscript",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $n\\geq 3$ be an integer and $p$ be a prime with $p\\equiv 1\\pmod{n}$. In this paper, we show that $${}_nF_{n-1}\\bigg[\\begin{matrix} \\frac{n-1}{n}&\\frac{n-1}{n}&\\ldots&\\frac{n-1}{n} &1&\\ldots&1\\end{matrix}\\bigg | \\, 1\\bigg]_{p-1}\\equiv -\\Gamma_p\\bigg(\\frac{1}{n}\\bigg)^n\\pmod{p^3}, $$ where the truncated hypergeometric series $$_nF_{n-1}\\bigg[\\begin{matrix} x_1&x_2&\\ldots&x_n &y_1&\\cdots&y_{n-1}\\end{matrix}\\bigg | \\, z\\bigg]_m=\\sum_{k=0}^{m}\\frac{z^k}{k!}\\prod_{j=0}^{k-1}\\frac{(x_1+j)\\cdots(x_{n}+j)}{(y_1+j)\\cdots(y_{n-1}+j)} $$ and $\\Gamma_p$ denotes the $p$-adic gamma function. This confirms a conjecture of Deines, Fuselier, Long, Swisher and Tu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-07T15:43:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C20",
      "05A10",
      "11B65",
      "11A07",
      "33E50"
    ],
    "keywords": [
      "supercongruence concerning truncated hypergeometric series",
      "adic gamma function",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}