Birth-death processes and associated polynomials

We consider birth-death processes on the nonnegative integers and the corresponding sequences of orthogonal polynomials called birth-death polynomials. The sequence of associated polynomials linked with a sequence of birth-death polynomials and its orthogonalizing measure can be used in the analysis of the underlying birth-death process in several ways. We briefly review the known applications of associated polynomials, which concern transition and first-entrance time probabilities, and establish some new results in this vein. In particular, our findings indicate how the prevalence of recurrence or $\alpha$-recurrence in a birth-death process can be recognized from certain properties of the orthogonalizing measure for the associated polynomial