{
  "id": "1707.02417",
  "version": "v1",
  "published": "2017-07-08T09:10:18.000Z",
  "updated": "2017-07-08T09:10:18.000Z",
  "title": "On the derivatives $\\partial^{2}P_ν(z)/\\partialν^{2}$ and $\\partial Q_ν(z)/\\partialν$ of the Legendre functions with respect to their degrees",
  "authors": [
    "Radosław Szmytkowski"
  ],
  "comment": "15 pages",
  "journal": "Integral Transforms Spec. Funct. 28 (2017) 645-62",
  "doi": "10.1080/10652469.2017.1339039",
  "categories": [
    "math.CA"
  ],
  "abstract": "We provide closed-form expressions for the degree-derivatives $[\\partial^{2}P_{\\nu}(z)/\\partial\\nu^{2}]_{\\nu=n}$ and $[\\partial Q_{\\nu}(z)/\\partial\\nu]_{\\nu=n}$, with $z\\in\\mathbb{C}$ and $n\\in\\mathbb{N}_{0}$, where $P_{\\nu}(z)$ and $Q_{\\nu}(z)$ are the Legendre functions of the first and the second kind, respectively. For $[\\partial^{2}P_{\\nu}(z)/\\partial\\nu^{2}]_{\\nu=n}$, we find that $$\\displaystyle\\frac{\\partial^{2}P_{\\nu}(z)}{\\partial\\nu^{2}}\\bigg|_{\\nu=n}=-2P_{n}(z)\\textrm{Li}_{2}\\frac{1-z}{2}+B_{n}(z)\\ln\\frac{z+1}{2}+C_{n}(z),$$ where $\\textrm{Li}_{2}[(1-z)/2]$ is the dilogarithm function, $P_{n}(z)$ is the Legendre polynomial, while $B_{n}(z)$ and $C_{n}(z)$ are certain polynomials in $z$ of degree $n$. For $[\\partial Q_{\\nu}(z)/\\partial\\nu]_{\\nu=n}$ and $z\\in\\mathbb{C}\\setminus[-1,1]$, we derive $$\\displaystyle \\frac{\\partial Q_{\\nu}(z)}{\\partial\\nu}\\bigg|_{\\nu=n}=-P_{n}(z)\\textrm{Li}_{2}\\frac{1-z}{2}-\\frac{1}{2}P_{n}(z)\\ln\\frac{z+1}{2}\\ln\\frac{z-1}{2} +\\frac{1}{4}B_{n}(z)\\ln\\frac{z+1}{2}-\\frac{(-1)^{n}}{4}B_{n}(-z)\\ln\\frac{z-1}{2}-\\frac{\\pi^{2}}{6}P_{n}(z) +\\frac{1}{4}C_{n}(z)-\\frac{(-1)^{n}}{4}C_{n}(-z).$$ A counterpart expression for $[\\partial Q_{\\nu}(x)/\\partial\\nu]_{\\nu=n}$, applicable when $x\\in(-1,1)$, is also presented. Explicit representations of the polynomials $B_{n}(z)$ and $C_{n}(z)$ as linear combinations of the Legendre polynomials are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-08T09:10:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05",
      "33B30"
    ],
    "keywords": [
      "legendre functions",
      "legendre polynomial",
      "dilogarithm function",
      "closed-form expressions",
      "second kind"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Taylor-Francis"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}