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We consider Poisson's equation on the $n$-dimensional sphere in the situationwhere the inhomogeneous term has zero integral. Using a number of classical andmodern hypergeometric identities, we integrate this equation to produce theform of the fundamental solutions for any number of dimensions in terms ofgeneralised hypergeometric functions, with different closed forms for even andodd-dimensional cases.

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