Special function representations of the Poisson kernel on hyperbolic spaces

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.