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We prove and collect numerous explicit and computable results for thefractional Laplacian $(-\Delta)^s f(x)$ with $s>0$ as well as its whole spaceinverse, the Riesz potential, $(-\Delta)^{-s}f(x)$ with$s\in\left(0,\frac{1}{2}\right)$. Choices of $f(x)$ include weighted classicalorthogonal polynomials such as the Legendre, Chebyshev, Jacobi, Laguerre andHermite polynomials, or first and second kind Bessel functions with or withoutsinusoid weights. Some higher dimensional fractional Laplacians and Rieszpotentials of generalized Zernike polynomials on the unit ball and itscomplement as well as whole space generalized Laguerre polynomials are alsodiscussed. The aim of this paper is to aid in the continued development ofnumerical methods for problems involving the fractional Laplacian or the Rieszpotential in bounded and unbounded domains -- both directly by providing usefulbasis or frame functions for spectral method approaches and indirectly byproviding accessible ways to construct computable toy problems on which to testnew numerical methods.

Explicit fractional Laplacians and Riesz potentials of classical functions