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The questions basically aims at discussing the relative power of using probability generating functions, moment generating functions and characteristic functions as an example for Galton-Watson-processes (GWP).

Within the theory of GWP I see people extensively use the probability generating function $\mathbb E[s^X], x\in [0,1]$ for a random variable $X$.

So far, I have thought that the characteristic function $\mathbb E[e^{itX}]$ is the most powerful tool out of three mentioned above, because it yields all information about the distribution and has nice analytic properties, in particular always exists.

Is there a good reason why the probability generating function is used for GWP?

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All three are strictly equivalent, of course, hence each is exactly as powerful as the two others, in particular the generating function is always analytic on $[0,1)$. Some natural properties of branching processes are best expressed using generating functions though, for example the probability of extinction $q$ solves $q=f(q)$ where $f(s)=E[s^X]$ for every $s$ in $[0,1]$ and $X$ denotes the number of descendants of each individual.

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