Consider a Galton-Watson process in which each individual reproduces independently of all others and has probability aj (j = 0, 1, …) of giving rise to j progeny in the following generation, and in which there is an independent immigration component where bj, (j = 0, 1, …) is the probability that j individuals enter the population at each generation. Defining Xn (n = 0, 1, …) to be the population size at the n- th generation, it is known that {Xn} defines a Markov chain on the non-negative integers.