I want to find the PGF of $Z_t$ where $Z_t$ is the number of individuals in generation $t$ (in Galton-Watson process).
The offspring distribution for this Galton-Watson tree is given by $X \overset{d}{=}Geometric(p)$, i.e. $Pr(X = k) = p(1-p)^{k}$ for $k = 0,1,2,\dots$.
How can I find $G_{Z_t}(s)$ under the above conditions?
Any hint or solutions are more than welcome!
The Galton-Watson property gives you the recurrence $$ G_{Z_{t+1}}(s) = \varphi(G_{Z_t}(s)), \quad t \geq 0, $$ where $\varphi$ is the generating function of the offspring distribution, that is $$ \varphi(s) = \frac{p}{1 - (1-p)s}. $$ But after this, there is not much to do except say that $$ G_{Z_t}(s) = \varphi^n(G_{Z_0}(s)), $$ where $\varphi^n$ is the composition of $\varphi$ by itself $n$ times. I do not see a case where this last expression could simplify nicely.