{
  "id": "1407.8465",
  "version": "v5",
  "published": "2014-07-31T15:53:32.000Z",
  "updated": "2024-01-09T16:30:43.000Z",
  "title": "New congruences involving harmonic numbers",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "32 pages, final published version",
  "journal": "Nanjing Univ. J. Math. Biquarterly 40 (2023), 1--33",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k,\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)},\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\\sum_{0<j\\le k}1/j$ and $H_k^{(2)}=\\sum_{0<j\\le k}1/j^2$. In particular, we show that \\begin{gather*}\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k\\equiv(-1)^{\\langle a\\rangle_p}\\,2\\left(B_{p-1}(a)-B_{p-1}\\right)\\pmod p, \\\\\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)}\\equiv -E_{p-3}(a)\\pmod p, \\\\(2a-1)\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}\\equiv B_{p-2}(a)\\pmod p, \\end{gather*} where $\\langle a\\rangle_p$ stands for the least nonnegative integer $r$ with $a\\equiv r\\pmod{p}$, and $B_n(x)$ and $E_n(x)$ denote the Bernoulli polynomial of degree $n$ and the Euler polynomial of degree $n$ respectively. We also pose some new conjectures on congruences.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-08-11T15:57:05.000Z",
      "abstract": "Let $p>3$ be a prime. For any $p$-adic integer $a$, we determine $$\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k,\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)},\\ \\ \\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}$$ modulo $p^2$, where $H_k=\\sum_{0<j\\le k}1/j$ and $H_k^{(2)}=\\sum_{0<j\\le k}1/j^2$. In particular, we show that \\begin{gather*}\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k\\equiv(-1)^{\\langle a\\rangle_p}2\\left(B_{p-1}(a)-B_{p-1}\\right)\\pmod p, \\\\\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}kH_k^{(2)}\\equiv -E_{p-3}(a)\\pmod p, \\\\(2a-1)\\sum_{k=0}^{p-1}\\binom{-a}k\\binom{a-1}k\\frac{H_k^{(2)}}{2k+1}\\equiv B_{p-2}(a)\\pmod p, \\end{gather*} where $\\langle a\\rangle_p$ stands for the least nonnegative integer $r$ with $a\\equiv r\\pmod{p}$, and $B_n(x)$ and $E_n(x)$ denote the Bernoulli polynomial of degree $n$ and the Euler polynomial of degree $n$ respectively.",
      "comment": "22 pages. The current Th. 1.1, Cor. 1.1 and Section 2 are new",
      "journal": null,
      "doi": null
    },
    {
      "version": "v5",
      "updated": "2024-01-09T16:30:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "05A10",
      "11B68",
      "11B75"
    ],
    "keywords": [
      "harmonic numbers",
      "congruences",
      "adic integer",
      "bernoulli polynomial"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.8465S"
    }
  }
}