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


SIGMA 9 (2013), 042, 26 pages      arXiv:1209.6047      http://dx.doi.org/10.3842/SIGMA.2013.042

Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems

Howard S. Cohl
Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899-8910, USA

Received November 29, 2012, in final form May 28, 2013; Published online June 05, 2013

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.

Key words: fundamental solutions; polyharmonic equation; Jacobi polynomials; Gegenbauer polynomials; Chebyshev polynomials; eigenfunction expansions; separation of variables; addition theorems.

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