Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

TL;DR

This paper derives a fundamental solution for Laplace's equation on a hypersphere, expressed through various special functions, highlighting the symmetric antipodal nature in hyperspherical geometry.

Contribution

It provides the first explicit form of the antipodal fundamental solution of Laplace's equation on hyperspheres using multiple mathematical representations.

Findings

Explicit fundamental solution expressed via hypergeometric functions

Multiple equivalent integral and summation representations provided

Highlights symmetry properties in hyperspherical Laplace solutions

Abstract

Due to the isotropy of $d$-dimensional hyperspherical space, one expects there to exist a spherically symmetric opposite antipodal fundamental solution for its corresponding Laplace-Beltrami operator. The $R$-radius hypersphere ${\mathbf S}_R^d$ with $R>0$, represents a Riemannian manifold with positive-constant sectional curvature. We obtain a spherically symmetric opposite antipodal fundamental solution of Laplace's equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the trigonometric sine, finite summation expressions over trigonometric functions, Gauss hypergeometric functions, and in terms of the Ferrers function of the second with degree and order given by $d/2-1$ and $1-d/2$ respectively, with real argument $x\in(-1,1)$.