  On even-dimensional Euclidean space for integer powers of the Laplace
operator greater than or equal to half the dimension, a fundamental solution of
the polyharmonic equation has binomial and logarithmic behavior. Gegenbauer
polynomial expansions of these fundamental solutions are obtained through a
limit applied to Gegenbauer expansions of a power-law fundamental solution of
the polyharmonic equation. This limit is accomplished through parameter
differentiation. By combining these results with previously derived azimuthal
Fourier series expansions for these binomial and logarithmic fundamental
solutions, we are able to obtain addition theorems for the azimuthal Fourier
coefficients. These logarithmic and binomial addition theorems are expressed in
Vilenkin polyspherical geodesic polar coordinate systems and as well in
generalized Hopf coordinates on spheres in arbitrary even dimensions.
