  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.
