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

In a recent paper in the Journal of Theoretical Probability 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 $\mathbb{E}[e^{s{T}_{ij}}]$ of the first-hitting time $T_{ij}$ for $any$ pair of states $i$ and $j$, as well as asymptotics for $\mathbb{E}[e^{s{T}_{ij}}]$ 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 $Marko{v}^{\mathrm{\prime }}s$ $theorem$.