A note on a functional equation arising in Galton-Watson branching processes

The functional equation ϕ ( mu ) = h ( ϕ ( u )) where is a probability generating function with 1 < m = h' (1 –) < ∞ and where F ( t ) is a non-decreasing right continuous function with F (0 –) = 0, F (0 +) < 1 and F (+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.