### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 079, 20 pages      arXiv:1508.06689      https://doi.org/10.3842/SIGMA.2016.079
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

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

Richard Chapling
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, England

Received November 23, 2015, in final form August 04, 2016; Published online August 10, 2016

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.

Key words: hyperspherical geometry; fundamental solution; Laplace's equation; separation of variables; hypergeometric functions.

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