  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.
