Fourier and Gegenbauer expansions for fundamental solutions of the Laplacian and powers in Euclidean and hyperbolic space

__ __

__Howard Cohl__

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

I compute derivatives with respect to parameters of
associated Legendre functions. Relatively little is known about the properties
of associated Legendre functions in relation to differentiation with respect to
their degree and order. There are many applications in special function theory
where having expressions for the derivatives with respect to these variables
can be useful. I have computed closed-form expressions for some of these
derivatives. A complex generalization of Heine's reciprocal square root
identity is presented. My complex generalization of Heine's reciprocal square
root identity is extremely useful for computing Fourier expansions of fundamental
solutions for the Laplacian and powers of the Laplacian in Euclidean space
about angles in separable axisymmetric coordinate systems. In hyperspherical
coordinates, I computed Gegenbauer polynomial expansions for a fundamental
solution of the Laplacian in Euclidean space. Gegenbauer expansions are useful
in conjunction with the addition theorem for hyperspherical harmonics for
generating addition theorems for Fourier coefficients of a fundamental solution
about angles in pure hyperspherical coordinate systems. Addition theorems for
associated Legendre functions of the second kind have been computed. We
present one of several possible addition theorems for the Fourier coefficients
of a fundamental solution of the Laplace equation in standard hyperspherical
coordinates in Euclidean space. Standard hyperspherical coordinates are the
most natural generalization to *d*-dimensions of spherical coordinates in
3-dimensional Euclidean space. We also present fundamental solutions of the
Laplacian (Laplace-Beltrami operator) in *d*-dimensional hyperbolic
geometry.