A Hypergeometric Integral with Applications to the Fundamental Solution of Laplace's Equation on Hyperspheres

TL;DR

This paper derives explicit fundamental solutions for Poisson's equation on n-dimensional spheres using hypergeometric functions, providing formulas applicable to any dimension with distinctions for even and odd cases.

Contribution

It introduces a novel approach to solving Poisson's equation on hyperspheres by expressing solutions in terms of hypergeometric functions for all dimensions.

Findings

Explicit formulas for fundamental solutions in all dimensions

Different closed forms for even and odd dimensions

Application of hypergeometric identities to PDE solutions

Abstract

We consider Poisson's equation on the $n$-dimensional sphere in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, we integrate this equation to produce the form of the fundamental solutions for any number of dimensions in terms of generalised hypergeometric functions, with different closed forms for even and odd-dimensional cases.