  By providing instances of approximation of linear diffusions by birth-death
processes, Feller [13], has offered an original path from the discrete world to
the continuous one. In this paper, by identifying an intertwining relationship
between squared Bessel processes and some linear birth-death processes, we show
that this connection is in fact more intimate and goes in the two directions.
As by-products, we identify some properties enjoyed by the birth-death family
that are inherited from squared Bessel processes. For instance, these include a
discrete self-similarity property and a discrete analogue of the beta-gamma
algebra. We proceed by explaining that the same gateway identity also holds for
the corresponding ergodic Laguerre semi-groups. It follows again that the
continuous and discrete versions are more closely related than thought before,
and this enables to pass information from one semi-group to the other one.
