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

2012/09/30 by Howard S. Cohl
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #Chebyshev polynomials #Classical orthogonal polynomials #Euclidean space #Fourier series #Fourier transform #Gegenbauer polynomials #Invariant (physics) #Jacobi polynomials #Mathematical analysis #Mathematical physics #Mathematics #Nonlinear Waves and Solitons #Orthogonal polynomials #Polynomial #Pure mathematics #Quantum Mechanics and Non-Hermitian Physics #math-ph #math.AP #math.CA #math.MP #msc:31B30 #msc:31C12 #msc:33C05 #msc:35A08 #msc:42A16

paper · pdf · doi:10.3842/sigma.2013.042

published as SIGMA 9 (2013), 042, 26 pages

arxiv created 2013/06/05 · openalex publication_date 2013/06/05 · arxiv updated 2013/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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.

Citations

• Formulas and Theorems for the Special Functions of Mathematical Physics
• One-loop corrections to the mass of self-dual semi-local planar topological solitons
• Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry
• Classical and Quantum Orthogonal Polynomials in One Variable