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.

翻译：本文证明并收集了分数阶拉普拉斯算子 $(-\Delta)^s f(x)$（其中 $s>0$）及其全空间逆算子——里斯势 $(-\Delta)^{-s}f(x)$（其中 $s\in\left(0,\frac{1}{2}\right)$）的诸多显式可计算结果。函数 $f(x)$ 的选择包括加权经典正交多项式，如勒让德、切比雪夫、雅可比、拉盖尔和埃尔米特多项式，以及带或不带正弦权重的第一类和第二类贝塞尔函数。此外，还讨论了单位球及其补空间上广义泽尼克多项式以及全空间广义拉盖尔多项式的高维分数阶拉普拉斯算子与里斯势。本文旨在促进涉及有界与无界域中分数阶拉普拉斯算子或里斯势的数值方法的发展——既通过为谱方法提供有用的基函数或框架函数直接助力，也通过构建可计算的示例问题间接地为新数值方法的测试提供便捷途径。