Fractional Laplace operator and Meijer G-function

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 harmoni...

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Bibliographic Details
Published in:arXiv.org 2015-09
Main Authors: Dyda, Bartłomiej, Kuznetsov, Alexey, Kwaśnicki, Mateusz
Format: Article
Language:English
Subjects:
Boundary conditions
Dirichlet problem
Eigenvalues
Eigenvectors
Hypergeometric functions
Laplace transforms
Nonlinear programming
Operators (mathematics)
Polynomials
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Summary: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".
ISSN:2331-8422