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.