{
  "id": "math/0608328",
  "version": "v1",
  "published": "2006-08-14T04:07:22.000Z",
  "updated": "2006-08-14T04:07:22.000Z",
  "title": "Fleck quotients and Bernoulli numbers",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "38 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Let p be a prime, and let n>0 and r be integers. In 1913 Fleck showed that $$F_p(n,r)=(-p)^{-[(n-1)/(p-1)]}\\sum_{k=r(mod p)}\\binom{n}{k}(-1)^k\\in\\Z.$$ Nowadays this result plays important roles in many aspects. Recently Sun and Wan investigated $F_p(n,r)$ mod p in [SW2]. In this paper, using p-adic methods we determine $(F_p(m,r)-F_p(n,r))/(m-n)$ modulo p in terms of Bernoulli numbers, where m>0 is an integer with $m\\not=n$ and $m=n (mod p(p-1))$. Consequently, $F_p(n,r)$ mod $p^{ord_p(n)+1}$ is determined; for example, if $n=n_*(mod p-1)$ with $0<n_*<p-2$ then $$\\frac{F_p(pn,0)}{pn}=\\frac{n_*!}{n_*+1}B_{p-1-n_*} (mod p).$$ This yields an application to Stirling numbers of the second kind. We also study extended Fleck quotients; in particular we prove that if $a>0$ and $l\\ge 0$ are integers with $2\\le n-l\\le p$ then $$\\frac{1}{p^{n-l}}\\sum_{l<k\\le n} \\binom{p^a n-d}{p^a k-d}(-1)^{pk}\\binom{k-1}{l} =\\frac{(-1)^{l-1}n!}{l!(n-l)}B_{p-n+l} (mod p)$$ for all d=1,...,max{p^{a-2},1}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-08-14T04:07:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A10",
      "11A07",
      "11B68",
      "11B73",
      "11S80",
      "11T24"
    ],
    "keywords": [
      "bernoulli numbers",
      "result plays important roles",
      "study extended fleck quotients",
      "second kind",
      "p-adic methods"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......8328S"
    }
  }
}