Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

TL;DR

This paper derives Fourier and Gegenbauer expansions for the fundamental solution of Laplace's equation on the hyperboloid model of hyperbolic geometry, providing explicit formulas and an addition theorem.

Contribution

It introduces explicit Fourier and Gegenbauer expansions for fundamental solutions in hyperbolic space, including an addition theorem in three dimensions.

Findings

Explicit Fourier expansions for hyperbolic fundamental solutions

Gegenbauer polynomial expansion in geodesic polar coordinates

Addition theorem for Fourier coefficients in 3D hyperbolic space

Abstract

Due to the isotropy $d$-dimensional hyperbolic space, there exist a spherically symmetric fundamental solution for its corresponding Laplace-Beltrami operator. On the $R$-radius hyperboloid model of $d$-dimensional hyperbolic geometry with $R>0$ and $d\ge 2$, we compute azimuthal Fourier expansions for a fundamental solution of Laplace's equation. For $d\ge 2$, we compute a Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace's equation on this negative-constant sectional curvature Riemannian manifold. In three-dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace's equation is obtained through comparison with its corresponding Gegenbauer expansion.