A hypergeometric integral with applications to the fundamental solution of Laplace's equation on hyperspheres (Q308689)

In the paper a Poisson's equation on the \(n\)-dimensional sphere is examined in the situation where the inhomogeneous term has zero integral. Using a number of classical and modern hypergeometric identities, the author integrates 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.NEWLINENEWLINEThe even- and odd-dimensional cases are treated separately: one may be expressed as a hypergeometric polynomial in \(\cot \frac{1}{2}\theta\), the other as a sum of an even polynomial in \(\cot \theta\) and an odd polynomial multiplied by \(x-\theta\). There is reason to expect this disparity, given the other ways in which spheres of even and odd dimension are different, such as the hairy ball theorem and the volume in terms of factorials.NEWLINENEWLINEThe last section gives some applications of the result including, among others, the dipole potential on the sphere and the azimuthal Fourier expansion.