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 α-recurrence in a birth–death process can be recognized from certain properties of the orthogonalizing measure for the associated polynomials.