Fractional Laplace operator and Meijer G-function

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".