{
  "id": "1407.0967",
  "version": "v5",
  "published": "2014-07-03T15:58:17.000Z",
  "updated": "2014-12-23T17:58:39.000Z",
  "title": "Congruences involving $g_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{2k}kx^k$",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "23 pages. Refined version. Th. 2.4 is new",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Define $g_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{2k}kx^k$ for $n=0,1,2,\\ldots$. Those numbers $g_n=g_n(1)$ are closely related to Ap\\'ery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for any prime $p>5$ we have $$\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k}\\equiv 0\\pmod{p^2}\\quad\\mbox{and}\\quad\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k^2}\\equiv 0\\pmod p.$$ This is similar to Wolstenholme's classical congruences $$\\sum_{k=1}^{p-1}\\frac1k\\equiv0\\pmod{p^2}\\quad\\mbox{and}\\quad\\sum_{k=1}^{p-1}\\frac{1}{k^2}\\equiv0\\pmod p$$ for any prime $p>3$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-07-23T19:58:50.000Z",
      "abstract": "Define $g_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{2k}kx^k$ for $n=0,1,2,\\ldots$. Those numbers $g_n=g_n(1)$ are closely related to Ap\\'ery numbers and Franel numbers. In this paper we establish some fundamental congruences involving $g_n(x)$. For example, for any prime $p>3$ we have $$\\sum_{k=1}^{p-1}g_k\\equiv0\\pmod{p^2}\\quad\\mbox{and}\\quad\\sum_{k=1}^{p-1}kg_k\\equiv -\\frac 34\\pmod{p^2}.$$ We also show the congruences $$\\sum_{k=1}^{p-1}\\frac{g_k(-1)}k\\equiv 0\\pmod{p^2}\\quad\\mbox{and}\\quad\\sum_{k=1}^{p-1}\\frac{g_k(-1)}{k^2}\\equiv 0\\pmod p$$ for any prime $p>5$.",
      "comment": "24 pages. Th. 2.1 is new",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2014-12-23T17:58:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "05A10",
      "05A30",
      "11B75"
    ],
    "keywords": [
      "fundamental congruences",
      "apery numbers",
      "franel numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.0967S"
    }
  }
}