{
  "id": "1407.8443",
  "version": "v1",
  "published": "2014-07-31T14:51:18.000Z",
  "updated": "2014-07-31T14:51:18.000Z",
  "title": "The 2-adic valuations of differences of Stirling numbers of the second kind",
  "authors": [
    "Wei Zhao",
    "Jianrong Zhao",
    "Shaofang Hong"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $m, n, k$ and $c$ be positive integers. Let $\\nu_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if $2\\le m\\le n$ and $c$ is odd, then $\\nu_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $\\nu_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-31T14:51:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "second kind",
      "stirling numbers",
      "differences",
      "convolution identity",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.8443Z"
    }
  }
}