{
  "id": "1107.2680",
  "version": "v2",
  "published": "2011-07-13T21:39:22.000Z",
  "updated": "2011-11-14T17:27:14.000Z",
  "title": "Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut (-1,1)",
  "authors": [
    "Radosław Szmytkowski"
  ],
  "comment": "LaTeX2e, 5 pages, some corrections and improvements made",
  "categories": [
    "math.CA",
    "math-ph",
    "math.CV",
    "math.MP"
  ],
  "abstract": "We use the recent findings of Cohl [arXiv:1105.2735] and evaluate two integrals involving the Gegenbauer polynomials: $\\int_{-1}^{x}\\mathrm{d}t\\:(1-t^{2})^{\\lambda-1/2}(x-t)^{-\\kappa-1/2}C_{n}^{\\lambda}(t)$ and $\\int_{x}^{1}\\mathrm{d}t\\:(1-t^{2})^{\\lambda-1/2}(t-x)^{-\\kappa-1/2}C_{n}^{\\lambda}(t)$, both with $\\Real\\lambda>-1/2$, $\\Real\\kappa<1/2$, $-1<x<1$. The results are expressed in terms of the on-the-cut associated Legendre functions $P_{n+\\lambda-1/2}^{\\kappa-\\lambda}(\\pm x)$ and $Q_{n+\\lambda-1/2}^{\\kappa-\\lambda}(x)$. In addition, we find closed-form representations of the series $\\sum_{n=0}^{\\infty}(\\pm)^{n}[(n+\\lambda)/\\lambda]P_{n+\\lambda-1/2}^{\\kappa-\\lambda}(\\pm x)C_{n}^{\\lambda}(t)$ and $\\sum_{n=0}^{\\infty}(\\pm)^{n}[(n+\\lambda)/\\lambda]Q_{n+\\lambda-1/2}^{\\kappa-\\lambda}(\\pm x)C_{n}^{\\lambda}(t)$, both with $\\Real\\lambda>-1/2$, $\\Real\\kappa<1/2$, $-1<t<1$, $-1<x<1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-11-14T17:27:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C55",
      "33C45",
      "33C05"
    ],
    "keywords": [
      "gegenbauer polynomials",
      "on-the-cut associated legendre functions",
      "closed-form representations"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.2680S"
    }
  }
}