  Let $(X_t)_{t\geq0}$ be the $n$-dimensional hyperbolic Brownian motion, that
is the diffusion on the real hyperbolic space $\D^n$ having the
Laplace-Beltrami operator as its generator. The aim of the paper is to derive
the formulas for the Gegenbauer transform of the Poisson kernel and the Green
function of the ball for the process $(X_t)_{t\geq0}$. Under some additional
hypotheses we give the formulas for the Poisson kernel itself. In particular,
we provide formulas in $\D^4$ and $\D^6$ spaces for the Poisson kernel and the
Green function as well.
