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Fourier and Gegenbauer expansions for a fundamental solution of Laplace’s equation in hyperspherical geometry. (English) Zbl 1311.31005

Summary: For a fundamental solution of Laplace’s equation on the \(R\)-radius \(d\)-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace’s equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace’s equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson’s equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.

MSC:

31C12 Potential theory on Riemannian manifolds and other spaces
33C05 Classical hypergeometric functions, \({}_2F_1\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C55 Spherical harmonics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35A08 Fundamental solutions to PDEs
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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