Laplace transforms, Mellin transforms and mixed poisson processes

While trying to generalize results of McNolty (1964), the present author discovered the usefulness of Laplace and Mellin transforms for the computation of the counting distribution Pn (t)=P(N(t)=n) of a mixed Poisson process {N(t); t⩾0}. For various mixing distributions U we present the corresponding expressions for Pn (t), expressions which have not been contained in literature so far, but also neater forms or easier derivations of existing results. Mixing distributions which are considered are: χ 2, χ, Maxwell, Rayleigh, Weibull, generalized gamma, Pearson Type I, III, V, VI, F, Pareto, Bessel and truncated normal.