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On even-dimensional Euclidean space for integer powers of the Laplaceoperator greater than or equal to half the dimension, a fundamental solution ofthe polyharmonic equation has binomial and logarithmic behavior. Gegenbauerpolynomial expansions of these fundamental solutions are obtained through alimit applied to Gegenbauer expansions of a power-law fundamental solution ofthe polyharmonic equation. This limit is accomplished through parameterdifferentiation. By combining these results with previously derived azimuthalFourier series expansions for these binomial and logarithmic fundamentalsolutions, we are able to obtain addition theorems for the azimuthal Fouriercoefficients. These logarithmic and binomial addition theorems are expressed inVilenkin polyspherical geodesic polar coordinate systems and as well ingeneralized Hopf coordinates on spheres in arbitrary even dimensions.

Gegenbauer expansions and addition theorems for a binomial and  logarithmic fundamental solution of the even-dimensional Euclidean  polyharmonic equation