{
  "id": "2202.05077",
  "version": "v1",
  "published": "2022-02-10T15:01:36.000Z",
  "updated": "2022-02-10T15:01:36.000Z",
  "title": "Congruences for sums involving products of three binomial coefficients",
  "authors": [
    "Zhi-Hong Sun"
  ],
  "comment": "50 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let $p>3$ be a prime, and let $a$ be a rational $p$-adic integer, using WZ method we establish the congruences modulo $p^3$ for $$\\sum_{k=0}^{p-1} \\binom ak\\binom{-1-a}k\\binom{2k}k\\frac {w(k)}{4^k},$$ where $$w(k)=1,\\frac 1{k+1},\\frac 1{(k+1)^2},\\frac 1{(k+1)^3},\\frac 1{2k-1},\\frac 1{k+2}, k,k^2,k^3,\\frac 1{a+k},\\frac 1{a+k-1}.$$ As consequences, taking $a=-\\frac 12,-\\frac 13,-\\frac 14,-\\frac 16$ we deduce many congruences modulo $p^3$ and so solve some conjectures posed by the author earlier.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-02-10T15:01:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "05A19",
      "11B65",
      "11B68",
      "11E25"
    ],
    "keywords": [
      "binomial coefficients",
      "congruences modulo",
      "wz method",
      "author earlier",
      "adic integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}