The p.g.f, $f_X(s)$ of a random variable $X$ is given by
$$ f_X(s) = \sum_{x = 0}^\infty P(X = x)s^x$$
Usually, it's said the $s$ is just a dummy variable, but having been exposed to the result that the smallest root of $f(s) = s$ is the extinction probability of a branching process, it's clear that there is meaning to $s$, but I don't understand it's general meaning.
Any further intuition is highly appreciated.