Special function representations of the Poisson kernel on hyperbolic spaces

Abstract

In this paper we compute explicit formulae for the Poisson kernels on the hyperbolic upper half-space $\mathbf {H}^{n}$ and the Poincaré unit ball $\mathbf {D}^{n}$. We first construct an associated Legendre function expression for eigenfunctions of the Laplacian and use superposition principle to get a solution for the Laplace equation on $\mathbf {H}^{n}$. The Poisson kernel on $\mathbf {D}^{n}$ is obtained from that on $\mathbf {H}^{n}$ by letting the hyperbolic distance $\rho =d(w,w')$ $(w,w'\in \mathbf {H}^{n})$ tend to infinity. These Poisson kernels, apart from being interesting in their own right lead to various identities that seem to be novel in the context of special functions.