Transition density of a hyperbolic Bessel process

We investigate the transition density of a hyperbolic Bessel process for integer dimensions and show a link between transition densities of a hyperbolic Brownian motion and a Bessel process in the same dimension. Using the so-called Millson's formula for the densities of hyperbolic Brownian motion we also show a link between transition density of n-dimensional hyperbolic Bessel process and 2-dimensional (if n is even) or 3-dimensional (if n is odd) hyperbolic Brownian motion. This helps us to get explicit formulas for the Bessel process transition density.