  We show that the multiplication operator associated to a fractional power of
a Gamma random variable, with parameter q>0, maps the convex cone of the
1-invariant functions for a self-similar semigroup into the convex cone of the
q-invariant functions for the associated Ornstein-Uhlenbeck (for short OU)
semigroup. We also describe the harmonic functions for some other
generalizations of the OU semigroup. Among the various applications, we
characterize, through their Laplace transforms, the laws of first passage times
above and overshoot for certain two-sided stable OU processes and also for
spectrally negative semi-stable OU processes. These Laplace transforms are
expressed in terms of a new family of power series which includes the
generalized Mittag-Leffler functions.
