Some applications of the Mellin transform to branching processes

We introduce a Mellin transform of functions which live on all of $\text{bR}$ and discuss its applications to the limiting theory of Bellman-Harris processes, and specifically Luria-Delbr"uck processes. More precisely, we calculate the life-time distribution of particles in a Bellman-Harris process from their first-generation offspring and limiting distributions, and prove a formula for the Laplace transform of the distribution of types in a Luria-Delbr"uck process in the Mittag-Leffler limit. As a by-product, we show how to easily derive the (classical) Mellin transforms of certain stable probability distributions from their Fourier transform.