Fractional Laplace Operator and Meijer G-function

• Published in: Constructive approximation Vol. 45; no. 3; pp. 427 - 448
• Main Authors: Dyda, Bartłomiej, Kuznetsov, Alexey, Kwaśnicki, Mateusz
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2017; Springer Nature B.V
• Subjects: Analysis; Boundary conditions; Eigenvalues; Eigenvectors; Hypergeometric functions; Laplace transforms; Mathematics; Mathematics and Statistics; Numerical Analysis; Operators (mathematics); Polynomials; Fractional Laplace operator; Radial function; 33C55; 33C20; Hypergeometric function; Jacobi polynomial; Meijer G-function; Harmonic polynomial; Riesz potential; 35S05
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-016-9336-4

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 ) + α / 2 ( - Δ ) α / 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 ).