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 (Dyda et al., Eigenvalues of the fractional Laplace operator in the unit ball, 2015, arXiv:1509.08533).