{
  "id": "2002.12072",
  "version": "v1",
  "published": "2020-02-27T13:02:50.000Z",
  "updated": "2020-02-27T13:02:50.000Z",
  "title": "Super congruences concerning binomial coefficients and Apéry-like numbers",
  "authors": [
    "Zhi-Hong Sun"
  ],
  "comment": "35 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime with $p>3$, and let $a,b$ be two rational $p-$integers. In this paper we present general congruences for $\\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\frac p{k+b}\\pmod {p^2}$. For $n=0,1,2,\\ldots$ let $D_n$ and $b_n$ be Domb and Almkvist-Zudilin numbers, respectively. We also establish congruences for $$\\sum_{n=0}^{p-1}\\frac{D_n}{16^n},\\quad \\sum_{n=0}^{p-1}\\frac{D_n}{4^n}, \\quad \\sum_{n=0}^{p-1}\\frac{b_n}{(-3)^n},\\quad \\sum_{n=0}^{p-1}\\frac{b_n}{(-27)^n}\\pmod {p^2}$$ in terms of certain binary quadratic forms.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-27T13:02:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "05A19",
      "11B65",
      "11E25"
    ],
    "keywords": [
      "super congruences concerning binomial coefficients",
      "apéry-like numbers",
      "binary quadratic forms",
      "almkvist-zudilin numbers",
      "general congruences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}