I am trying to work through a problem, but I am not sure if I am heading in the right direction. For a branching process $Z_n$, the total population ever born is given by $R = 1 + \sum_{n=1}^\infty Z_n$. I now must find a recursion that is satisfied by the generating function of $R$.
I have done the following:
$R = 1 + \sum_{n=1}^\infty Z_n = \sum_{n=0}^\infty Z_n$ with $Z_0 = 1$.
The generating function of a branching process is $P_n(s) = P(P_{n-1}(s))$, with $P_0(s) = s$ and $P_1(s) = P(s)$. Since we are adding over the entire population ever born, I am creating the PGF as
$\mathbb{E}[s^{Z_0 + Z_1 + ...}] = \mathbb{E}[s^{Z_0}] \mathbb{E}[s^{Z_1}]... = \prod_{i=0}^\infty P_{Z_i}(s)$,
$P_S(s) = \prod_{i=0}^\infty P_{Z_i}(s) = s \prod_{i=1}^\infty P_{Z_i}(s) = s \prod_{i=1}^\infty P(P_{i-1}(s))$.
For example, if I were to try to solve this with the PGF of a Bernoulli rv, $P(s) = q + ps$, how would I solve this? For a branching process with Bernoulli rv's, the process goes exinct as $n \rightarrow \infty$. But I am not sure how to incorporate this in the solution since it is a product, rather than a sum.
In first branch : $$S=1+\sum_{i=1}^{Z_1}S_i$$Where $S_i$ is the number of offspring(containing himself) of the ith population in the first generation. If $S$ has generate function $F(s)$,So are $S_i$s .So$$F(s)=sF_{Z_1}(F_{S_i}(s))=sP(F(s)).$$