{
  "id": "1603.03547",
  "version": "v1",
  "published": "2016-03-11T07:28:39.000Z",
  "updated": "2016-03-11T07:28:39.000Z",
  "title": "Two Definite Integrals Involving Products of Four Legendre Functions",
  "authors": [
    "Yajun Zhou"
  ],
  "comment": "12 pages. Proof of two integral formulae stated in arXiv:1301.1735v4 and arXiv:1506.00318v2",
  "categories": [
    "math.CA",
    "math.NT"
  ],
  "abstract": "The definite integrals $ \\int_{-1}^1x[P_\\nu(x)]^4\\,\\mathrm{d} x$ and $ \\int_{0}^1x[P_\\nu(x)]^2\\{[P_\\nu(x)]^2-[P_\\nu(-x)]^2\\}\\,\\mathrm{d} x$ are evaluated in closed form, where $ P_\\nu$ stands for the Legendre function of degree $ \\nu\\in\\mathbb C$. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green's functions and Epstein zeta functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-11T07:28:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05",
      "33C15",
      "11F03"
    ],
    "keywords": [
      "legendre function",
      "definite integrals",
      "epstein zeta functions",
      "automorphic greens functions",
      "special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160303547Z"
    }
  }
}