{
  "id": "1407.0636",
  "version": "v6",
  "published": "2014-07-02T16:24:45.000Z",
  "updated": "2014-07-28T18:08:48.000Z",
  "title": "Super congruences involving Bernoulli and Euler polynomials",
  "authors": [
    "Zhi-Hong Sun"
  ],
  "comment": "30 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\\{B_n(x)\\}$ and $\\{E_n(x)\\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\equiv (-1)^{\\langle a\\rangle_p}+ p^2t(t+1)E_{p-3}(-a)\\pmod{p^3}$$ and for $a\\not\\equiv -\\frac 12\\pmod p$, $$\\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\frac 1{2k+1}\\equiv \\frac{1+2t}{1+2a} +p^2\\frac{t(t+1)}{1+2a}B_{p-2}(-a)\\pmod{p^3},$$ where $\\langle a\\rangle_p\\in\\{0,1,\\ldots,p-1\\}$ satisfying $a\\equiv \\langle a\\rangle_p\\pmod p$ and $t=(a-\\langle a\\rangle_p)/p$. Taking $a=-\\frac 13,-\\frac 14,-\\frac 16$ in the above congruences we solve some conjectures of Z.W. Sun. In this paper we also establish congruences for $\\sum_{k=0}^{p-1}k\\binom ak\\binom{-1-a}k,\\ \\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\frac 1{2k-1},\\ \\sum_{k=1}^{p-1}\\frac 1k\\binom ak\\binom{-1-a}k\\pmod{p^3}$ and $\\sum_{k=1}^{p-1}\\frac {(-1)^k}k\\binom ak,\\ \\sum_{k=0}^{p-1}\\binom ak(-2)^k\\pmod{p^2}.$",
  "revisions": [
    {
      "version": "v6",
      "updated": "2014-07-28T18:08:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B68",
      "05A19"
    ],
    "keywords": [
      "euler polynomials",
      "super congruences",
      "rational p-adic integer",
      "bernoulli polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.0636S"
    }
  }
}