Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

TL;DR

This paper develops polynomial expansions for kernels related to power-law solutions of the polyharmonic equation in multiple coordinate systems, enabling new addition theorems for Fourier coefficients.

Contribution

It introduces new polynomial expansion methods for polyharmonic kernels and derives addition theorems in various coordinate systems, advancing mathematical tools for these equations.

Findings

Derived Fourier expansions in rotationally-invariant coordinates.

Established Gegenbauer polynomial expansions in polyspherical coordinates.

Generated addition theorems for azimuthal Fourier coefficients.

Abstract

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 certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients.