  In this paper we study explicit strong solutions for two
difference-differential fractional equations, defined via the generator of an
immigration-death process, by using spectral methods. Moreover, we give a
stochastic representation of the solutions of such difference-differential
equations by means of a stable time-changed immigration-death process and we
use this stochastic representation to show boundedness and then uniqueness of
these strong solutions. Finally, we study the limit distribution of the
time-changed process.
