{
  "id": "1412.0523",
  "version": "v1",
  "published": "2014-11-28T14:07:57.000Z",
  "updated": "2014-11-28T14:07:57.000Z",
  "title": "Two congruences involving harmonic numbers with applications",
  "authors": [
    "Guo-Shuai Mao",
    "Zhi-Wei Sun"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "The harmonic numbers $H_n=\\sum_{0<k\\le n}1/k\\ (n=0,1,2,\\ldots)$ play important roles in mathematics. Let $p>3$ be a prime. With helps of some combinatorial identities, we establish the following two new congruences: $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_k\\equiv\\frac13\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p}$$ and $$\\sum_{k=1}^{p-1}\\frac{\\binom{2k}k}kH_{2k}\\equiv\\frac7{12}\\left(\\frac p3\\right)B_{p-2}\\left(\\frac13\\right)\\pmod{p},$$ where $B_n(x)$ denotes the Bernoulli polynomial of degree $n$. As applications we determine $\\sum_{n=1}^{p-1}g_n$ and $\\sum_{n=1}^{p-1}h_n$ modulo $p^3$, where $$g_n=\\sum_{k=0}^n\\binom nk^2\\binom{2k}k\\quad\\mbox{and}\\quad h_n=\\sum_{k=0}^n\\binom nk^2C_k$$ with $C_k=\\binom{2k}k/(k+1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-28T14:07:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "11B68",
      "05A10",
      "11A07"
    ],
    "keywords": [
      "harmonic numbers",
      "congruences",
      "applications",
      "play important roles",
      "combinatorial identities"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.0523M"
    }
  }
}