On the Recognition and Structure of Probability Generating Functions

If $$ M\left( s \right) = 1 - {e^{ - \pi (s)}} $$ is a probability generating function, the coefficients π j in the MacLaurin expansion π(s) comprise a harmonic renewal sequence. A simple sufficient condition is given which ensures that a non-negative sequence is harmonic renewal. This condition covers the case of the limiting conditional law of a subcritical Markov branching process.Examples are given illustrating the limitation of the criterion. The parallel problem for continuous laws and its relation to the CB-process is discussed.