For a fundamental solution of Laplace's equation on the R-radius
d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in
closed form in two and three dimensions. We also compute the Gegenbauer
polynomial expansion for a fundamental solution of Laplace's equation in
hyperspherical geometry in geodesic polar coordinates. From this expansion in
three-dimensions, we derive an addition theorem for the azimuthal Fourier
coefficients of a fundamental solution of Laplace's equation on the 3-sphere.
Applications of our expansions are given, namely closed-form solutions to
Poisson's equation with uniform density source distributions. The Newtonian
potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular
curve segment on the 3-sphere. Applications are also given to the
superintegrable Kepler-Coulomb and isotropic oscillator potentials
