{
  "id": "1301.6586",
  "version": "v1",
  "published": "2013-01-28T16:04:28.000Z",
  "updated": "2013-01-28T16:04:28.000Z",
  "title": "The parameter derivatives $[\\partial^{2}P_ν(z)/\\partialν^{2}]_{ν=0}$ and $[\\partial^{3}P_ν(z)/\\partialν^{3}]_{ν=0}$, where $P_ν(z)$ is the Legendre function of the first kind",
  "authors": [
    "Radosław Szmytkowski"
  ],
  "comment": "5 pages",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We derive explicit expressions for the parameter derivatives $[\\partial^{2}P_{\\nu}(z)/\\partial\\nu^{2}]_{\\nu=0}$ and $[\\partial^{3}P_{\\nu}(z)/\\partial\\nu^{3}]_{\\nu=0}$, where $P_{\\nu}(z)$ is the Legendre function of the first kind. It is found that {displaymath} \\frac{\\partial^{2}P_{\\nu}(z)}{\\partial\\nu^{2}}\\bigg|_{\\nu=0} =-2\\Li_{2}\\frac{1-z}{2}, {displaymath} where $\\Li_{2}z$ is the dilogarithm (this formula has been recently arrived at by Schramkowski using \\emph{Mathematica}), and that {displaymath} \\frac{\\partial^{3}P_{\\nu}(z)}{\\partial\\nu^{3}}\\bigg|_{\\nu=0} =12\\Li_{3}\\frac{z+1}{2}-6\\ln\\frac{z+1}{2}\\Li_{2}\\frac{z+1}{2} -\\pi^{2}\\ln\\frac{z+1}{2}-12\\zeta(3), {displaymath} where $\\Li_{3}z$ is the polylogarithm of order 3 and $\\zeta(s)$ is the Riemann zeta function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-01-28T16:04:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05",
      "33B30"
    ],
    "keywords": [
      "legendre function",
      "first kind",
      "parameter derivatives",
      "displaymath",
      "riemann zeta function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1301.6586S"
    }
  }
}