{
  "id": "1209.1018",
  "version": "v2",
  "published": "2012-09-05T15:39:28.000Z",
  "updated": "2014-10-29T18:13:31.000Z",
  "title": "Identities between polynomials related to Stirling and harmonic numbers",
  "authors": [
    "Bernd C. Kellner"
  ],
  "comment": "20 pages; extended and final revised version",
  "journal": "Integers 14, (2014), Article A54, 1-22",
  "categories": [
    "math.NT"
  ],
  "abstract": "We consider two types of polynomials $F_n (x) = \\sum_{\\nu=1}^n \\nu! S_2(n,\\nu) x^\\nu$ and $\\hat{F}_n (x) = \\sum_{\\nu=1}^n \\nu! S_2(n,\\nu) H_\\nu x^\\nu$, where $S_2(n,\\nu)$ are the Stirling numbers of the second kind and $H_\\nu$ are the harmonic numbers. We show some properties and relations between these polynomials. Especially, the identity $\\hat{F}_n (-\\tfrac{1}{2}) = - (n-1)/2 \\cdot F_{n-1} (-\\tfrac{1}{2})$ is established for even $n$, where the values are connected with Genocchi numbers. For odd $n$ the value of $\\hat{F}_n (-\\tfrac{1}{2})$ is given by a convolution of these numbers. Subsequently, we discuss some of these convolutions, which are connected with Miki type convolutions of Bernoulli and Genocchi numbers, and derive some 2-adic valuations of them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-09-05T15:39:28.000Z",
      "title": "Identities between Polynomials Related to Stirling and Harmonic Numbers",
      "abstract": "We consider two types of polynomials $F_n(x) = \\sum_{\\nu=1}^n \\nu! S_2(n,\\nu) x^\\nu$ and $\\hat{F}_n(x) = \\sum_{\\nu=1}^n \\nu! S_2(n,\\nu) H_\\nu x^\\nu$ where $S_2(n,\\nu)$ are the Stirling numbers of the second kind and $H_\\nu$ are the harmonic numbers. We show some relations between these polynomials and an identity between the special values $\\hat{F}_{2n}(-1/2)$ and $F_{2n-1}(-1/2)$, which are connected with Bernoulli numbers.",
      "comment": "9 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-29T18:13:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B73",
      "11B83",
      "11B68"
    ],
    "keywords": [
      "harmonic numbers",
      "polynomials",
      "special values",
      "bernoulli numbers"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.1018K"
    }
  }
}