{
  "id": "1509.08529",
  "version": "v1",
  "published": "2015-09-28T22:40:10.000Z",
  "updated": "2015-09-28T22:40:10.000Z",
  "title": "Fractional Laplace operator and Meijer G-function",
  "authors": [
    "Bartłomiej Dyda",
    "Alexey Kuznetsov",
    "Mateusz Kwaśnicki"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x|^2, or generalized hypergeometric functions of -|x|^2, multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1-|x|^2)_+^{alpha/2} (-Delta)^{alpha/2} with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper \"Eigenvalues of the fractional Laplace operator in the unit ball\".",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-28T22:40:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unit ball",
      "laplace operator maps meijer g-functions",
      "fractional laplace operator maps meijer",
      "dirichlet boundary conditions outside",
      "solid harmonic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150908529D"
    }
  }
}