  In this paper we study strong solutions of some non-local
difference-differential equations linked to a class of birth-death processes
arising as discrete approximations of Pearson diffusions by means of a spectral
decomposition in terms of orthogonal polynomials and eigenfunctions of some
non-local derivatives. Moreover, we give a stochastic representation of such
solutions in terms of time-changed birth-death processes and study their
invariant and their limit distribution. Finally, we describe the correlation
structure of the aforementioned time-changed birth-death processes.
