=+13. In a subcritical branching process with immigration, let Q(s) be the progeny generating function and R(s) the generating function of the number of new immigrants at each generation. If the equilibrium distribution has generating function P∞(s), then show that P∞(s) = P∞(Q(s))R(s).

For the choices Q(s)=1 − p + ps and R(s) = e−λ(1−s), find P∞(s).

(Hint: Let P∞(s) be a Poisson generating function.)