I found this interesting question in Sidney Resnick's Adventurous Stochastic Processes, exercise 2.22 pp 154-155. I need help in solving the problem. The only similar one I found is here Branching Process with immigration - Finding pgf which has no answers for about 4years. I am sorry for not showing what I've done to solve the problem, I have limited background in stochastic. I hope my learned users here will pardoned me.
A Markov chain $\{Z_n\}$ with state space $\{0, 1, \ldots\}$ and transition matrix $P.$ Given two generating functions $Q(s)$ and $P(s)$ with $Q'(1) = \alpha <1,$ $P'(1) < \infty,$ related to the transition matrix by $$\sum_{j=0}^{\infty}p_{ij} s^j = P(s)Q(s)^i$$
a) Show that the transition probabilities corresponding to a branching process with immigration, namely a process constructed as follows. Let $\{Z_{n,j},n \geq 1, j\geq 1\}$ iid and non-negative random variables with a common generating function $Q(s).$ Let $\{I_n; n\geq1\}$ be iid, independent of $\{Z_{n,j}\},$ with generating function $P(s).$ Define $n \geq 0$ $$Z_{n+1} = I_{n+1} \sum_{j=1}^{Z_n} Z_{n,j}.$$ so that the random variable $I_{n}$ count the number of immigrants per generation.
b) show that $\{Z_n\}$ has a stationary distribution $\{\pi_k, k\geq0\}$ whose generating function $\Pi(s)$ satisfies $$\Pi(s) = P(s) \Pi (Q(s)).$$ Solve to get an expression for $\Pi(s)$ in terms of $Q$ and $P.$