Koenigs function and fractional iterates of probability generating functions

The Koenigs function arises as the limit of an appropriately normalized sequence of iterates of holomorphic functions. On the other hand it is a solution of a certain functional equation and can be used for the definition of iterates of the original function. A description of the class of Koenigs functions corresponding to probability generating functions embeddable in a one-parameter group of fractional iterates is provided. The results obtained can be regarded as a test for the embeddability of a Galton-Watson process in a homogeneous Markov branching process.