Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems

Published

June 5, 2013

Author(s)

Howard S. Cohl

Abstract

We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients.

Citation

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

Volume

9

Pub Type

Journals

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Keywords

fundamental solutions, polyharmonic equation, Jacobi polynomials, Gegenbauer polynomials, Chebyshev polynomials, eigenfunction expansions, separation of variables, addition theorems

Citation

Cohl, H. (2013), Fourier, Gegenbauer and Jacobi expansions for a power-law fundamental solution of the polyharmonic equation and polyspherical addition theorems, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=912042 (Accessed October 7, 2025)