LU-type factorizations for birth--death processes and their Darboux transformations

We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield new infinitesimal generators of birth--death processes. Two types are considered: lower--upper (LU), which is unique and upper--lower (UL), which involves a free parameter. For both cases, we determine the conditions under which such factorizations can occur, derive explicit formulas for their coefficients, and provide a probabilistic interpretation of the factors. The spectral properties and associated orthogonal polynomials of the Darboux transformations are also analyzed. Finally, the general results are applied to classical examples such as the $M/M/1$ and $M/M/\infty$ queues and to different cases of linear birth--death processes.