Any probability generating function $G(z)$ is analytic on at least the open unit disk of the complex plane, because its Taylor series expansion about $z=0$ has a radius of convergence of at least 1. Moreover, any PGF satisfies $G(1)=1$, so the Taylor series converges at that point.
But I don't know of any theorem that says that any PGF must be analytic in a neighborhood of $z=1$, or even that derivatives $G^{(n)}(1)$ must exist at $z=1$. In order to calculate the random variable's factorial moments, strictly speaking we must instead take the more cumbersome limit $\lim \limits_{z \to 1^-} G^{(n)}(z)$, which is guaranteed to exist by Abel's theorem.
Are there actually examples of PGFs that are not analytic on any neighborhood of $z=1$? (This could only happen if the PGF's radius of convergence equals 1.) The Wikipedia article on Abel's theorem linked above does give an example of a function $$f(z) = \sum_{n=1}^\infty \frac{z^{3^n}-z^{2 \cdot 3^n}}{n}$$ that is analytic on the open unit disk, and defined at $z=1$ but discontinuous there, so that no derivatives $f^{(n)}(1)$ exist for any positive $n$. But this function is not a probability generating function. Are there probability functions that are similarly pathological near $z=1$, so that we really do need to use Abel's theorem instead of just directly taking derivatives at $z=1$?