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

 

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).