Explicit fractional Laplacians and Riesz potentials of classical functions

TL;DR

This paper provides explicit formulas and computable results for fractional Laplacians and Riesz potentials of classical functions, aiding numerical methods development for fractional PDEs.

Contribution

It introduces explicit formulas for fractional Laplacians and Riesz potentials of classical orthogonal polynomials and special functions, facilitating numerical analysis.

Findings

Explicit formulas for fractional Laplacians of classical functions

Explicit formulas for Riesz potentials of classical functions

Extension to higher-dimensional polynomials and functions

Abstract

We prove and collect numerous explicit and computable results for the fractional Laplacian $(-\Delta)^s f(x)$ with $s>0$ as well as its whole space inverse, the Riesz potential, $(-\Delta)^{-s}f(x)$ with $s\in\left(0,\frac{1}{2}\right)$. Choices of $f(x)$ include weighted classical orthogonal polynomials such as the Legendre, Chebyshev, Jacobi, Laguerre and Hermite polynomials, or first and second kind Bessel functions with or without sinusoid weights. Some higher dimensional fractional Laplacians and Riesz potentials of generalized Zernike polynomials on the unit ball and its complement as well as whole space generalized Laguerre polynomials are also discussed. The aim of this paper is to aid in the continued development of numerical methods for problems involving the fractional Laplacian or the Riesz potential in bounded and unbounded domains -- both directly by providing useful basis or frame functions for spectral method approaches and indirectly by providing accessible ways to construct computable toy problems on which to test new numerical methods.