Fourier and Gegenbauer expansions for fundamental solutions of the Laplacian and powers in Rd and Hd

Abstract

We compute fundamental solutions and associated Fourier cosine series for the Laplacian (and its powers) in Euclidean space Rd and in hyperbolic space Hd by introducing natural coordinate systems. More specific, this is done as follows. We prove parameter derivative formulae for certain associated Legendre functions. We derive closed-form expressions of normalized fundamental solutions for powers of the Laplacian in Rd and compute Fourier expansions for these fundamental solutions in terms of natural angles in axisymmetric subgroup type coordinate systems. We give azimuthal and separation angle Fourier expansions for pure hyperspherical coordinate systems, as well as Fourier expansions in mixed Euclidean-hyperspherical coordinate systems. Using azimuthal Fourier expansions compared with Gegenbauer polynomial expansions of fundamental solutions for powers of the Laplacian, we construct multi-summation addition theorems in pure hyperspherical subgroup type coordinate systems. We give some examples of multi-summation addition theorems for a certain sub-class of pure hyperspherical coordinate systems in Rd for d ∈ {3, 4, . . .}. We also give an example of a logarithmic multi-summation addition theorem, namely that for an unnormalized fundamental solution for powers of the Laplacian in R4. In the d-dimensional hyperboloid model of hyperbolic geometry Hd, we compute spherically symmetric normalized fundamental solutions for the Laplace-Beltrami operator. Finally, we also compute Fourier expansions for unnormalized fundamental solutions for this space in two and three dimensions.