{
  "id": "1304.1606",
  "version": "v2",
  "published": "2013-04-05T03:20:21.000Z",
  "updated": "2013-05-24T18:38:49.000Z",
  "title": "On Some Integrals Over the Product of Three Legendre Functions",
  "authors": [
    "Yajun Zhou"
  ],
  "comment": "Revised according to referee's report, 9 pages",
  "doi": "10.1007/s11139-013-9504-0",
  "categories": [
    "math.CA"
  ],
  "abstract": "The definite integrals $ \\int_{-1}^1(1-x^2)^{(\\nu-1)/2}[P_\\nu(x)]^3\\D x$, $ \\int_{-1}^1(1-x^2)^{(\\nu-1)/2}[P_\\nu(x)]^2P_{\\nu}(-x)\\D x$, $\\int_{-1}^1x(1-x^2)^{(\\nu-1)/2}[P_{\\nu+1}(x)]^3\\D x $ and $\\int_{-1}^1x(1-x^2)^{(\\nu-1)/2}[P_{\\nu+1}(x)]^2P_{\\nu+1}(-x)\\D x $ are evaluated in closed form, where $P_\\nu$ is the Legendre function of degree $\\nu$, and $ \\R\\nu>-1$. Special cases of these formulae are related to certain integrals over elliptic integrals that have arithmetic interest.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-24T18:38:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05",
      "33C10",
      "33C45",
      "33C75",
      "33E05",
      "34E05",
      "34E13",
      "44A15"
    ],
    "keywords": [
      "legendre function",
      "definite integrals",
      "special cases",
      "elliptic integrals"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1304.1606Z"
    }
  }
}