Birth and death processes with absorption (Q1196602)

In a stationary birth and death process with birth rates \(\{\lambda_ n\}\) and death rates \(\{\mu_ n\}\), there is a long established spectral representation for the transition probabilities in the case where \(\lambda_ n>0\) for \(n\geq 0\) and \(\mu_ n>0\) for \(n\geq 1\), due to \textit{S. Karlin} and \textit{J. McGregor} [Trans. Am. Math. Soc. 85, 489-546 (1957)]. Here the case \(\lambda_ 0=0\) is covered by a careful study of what happens as \(\lambda_ 0\) tends to zero from above in the Karlin- McGregor representation. Some examples are discussed, and an inequality for Dirichlet polynomials is deduced.