1209.6047 (Howard S. Cohl)

Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental
solution of the polyharmonic equation and polyspherical addition theorems    [PDF]

Howard S. Cohl

We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on $d$-dimensional Euclidean space. From these series representations we derive Fourier expansions in rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. The resulting Fourier and Gegenbauer coefficients are given in terms of associated Legendre functions. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients.

View original: http://arxiv.org/abs/1209.6047