The probability-generating function of a random variable $X$ is $$ G(z) = E[z^X] $$
Assume that $X$ is a discrete random variable on $\{0,1,2,\ldots\}$ then $$ G(z) = \sum_{j = 0}^n z^j P(X = j) $$
I have always used the PGF when dealing with real $z$. I was curious if there are any interesting problems involving complex $z$ when $X$ is real.