  We consider an extended birth-death-immigration process defined on a lattice
formed by the integers of $d$ semiaxes joined at the origin. When the process
reaches the origin, then it may jumps toward any semiaxis with the same rate.
The dynamics on each ray evolves according to a one-dimensional linear
birth-death process with immigration. We investigate the transient and
asymptotic behavior of the process via its probability generating function. The
stationary distribution, when existing, is a zero-modified negative binomial
distribution. We also study a diffusive approximation of the process, which
involves a diffusion process with linear drift and infinitesimal variance on
each ray. It possesses a gamma-type transient density admitting a stationary
limit.
  As a byproduct of our study, we obtain a closed form of the number of
permutations with a fixed number of components, and a new series form of the
polylogarithm function expressed in terms of the Gauss hypergeometric function.
