An orthogonal-polynomial approach to first-hitting times of birth-death processes

In a recent paper in this journal, Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor’s classical results on first-hitting times of a birth–death process on the nonnegative integers by establishing a representation for the Laplace transform E[exp(sT_ij)] of the first-hitting time T_ij for any pair of states i and j, as well as asymptotics for E[exp(sT_ij)] when either i or j tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-polynomial toolbox used by Karlin and McGregor, in particular associated polynomials and Markov’s theorem.