THE SPECIAL FUNCTIONS OF FOURIER, GEGENBAUER, AND JACOBI ANALYSIS FOR FUNDAMENTAL SOLUTIONS OF THE POLYHARMONIC EQUATION ON HIGHLY SYMMETRIC MANIFOLDS

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__Howard Cohl__

Mathematician, Applied and Computational Mathematics, Mathematical Software Group, NIST

I perform harmonic analysis of fundamental solutions for the
Laplace and polyharmonic equations on highly symmetric (isotropic) manifolds.
These manifolds include Euclidean space, hyperbolic geometry and hyperspherical
geometry. In *d*-dimensional Euclidean space, a closed-form expression
for a fundamental solution of the polyharmonic equation is well-known. In
Euclidean space we perform Fourier, Gegenbauer, and Jacobi analysis of
power-law fundamental solutions of the polyharmonic equation and Fourier and Gegenbauer
analysis of logarithmic fundamental solutions of the polyharmonic equation.
Furthermore, we present simplifications of generating functions for Jacobi,
Gegenbauer, and Chebyshev polynomials in terms of associated Legendre
functions. In the hyperboloid model of *d*-dimensional hyperbolic
geometry, we use a fundamental solution of Laplace's equation to perform a
resulting Fourier and Gegenbauer analysis. We compare this analysis to
generate a single-summation addition theorem for the special function which
arises as the azimuthal Fourier coefficients. We determine a fundamental
solution of Laplace's equation on the *d*-dimensional hypersphere in terms
of a definite-integral, finite-summation expressions, and Ferrers functions of
the second kind (associated Legendre functions of the second kind on the cut).