Fractional iteration of probability generating functions and imbedding discrete branching processes in continuous processes

Fractional iteration of probability generating functions is investigated. In particular, conditions on the generating function of a Galton–Watson process that are necessary and sufficient for the process to admit imbedding in a continuous-time homogeneous Markov branching process are obtained. Necessary imbedding conditions formulated in terms of the several initial coefficients of the generating function are also obtained. The collection of all probability generating functions is partitioned, in accordance with a classification of branching processes, into subsets, and the latter are described as convex hulls of their extreme points. A description is given of the infinitesimal generators of distinguished semigroups of probability generating functions.