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

TL;DR

This paper derives Gegenbauer polynomial expansions and addition theorems for fundamental solutions of the polyharmonic equation in even-dimensional Euclidean spaces, revealing binomial and logarithmic behaviors for specific powers of the Laplacian.

Contribution

It introduces new Gegenbauer expansions and addition theorems for fundamental solutions with binomial and logarithmic behavior in even dimensions, extending previous Fourier series results.

Findings

Derived Gegenbauer polynomial expansions for fundamental solutions.

Established addition theorems in geodesic polar and Hopf coordinates.

Connected expansions with azimuthal Fourier series for these solutions.

Abstract

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.