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

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.

References in corpus (4)

• Contiguous relations of hypergeometric series
• Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry
• Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
• Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

Cited by in corpus (4)

• Opposite Antipodal Fundamental Solution of Laplace's Equation in Hyperspherical Geometry
• Fundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature
• Non-local scalar fields in static spacetimes via heat kernels
• Color confinement at the boundary of the conformally compactified ${\mathrm{AdS}}_5$