Numerical Approximation of the Fractional Laplacian on $\mathbb{R}$ Using Orthogonal Families

In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the ${}_2{F}_1$ Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the Higgins functions, the Christov functions, and their sine-like and cosine-like versions. After discussing the numerical difficulties in the implementation of the proposed formulas, we develop a method using variable precision arithmetic that gives accurate results.