Assume we have a random variable $X:(\Omega,\mathcal{F})\rightarrow (\mathbb{N}_\infty, \mathcal{P}(\mathbb{N}_\infty))$ with $\mathbb{N}_\infty=\mathbb{N}\cup(\left\{\infty\right\})$ with distribution $\mathbb{P}_X$ for a probability measure $\mathbb{P}$ on $\mathcal{F}$. Usually we define for a discrete r.v. the probability generating function $\pi$ as:
$\pi(z)=\mathbb{E}[z^X]= \sum_{k=1}^{\infty}z^k\cdot \mathbb{P}_X(k)$ for $z\in [0,1]$.
How can we extend this notion to cases where we expect even infinite values of $X$ with positive probability?
For example if we set $X=\infty$, then we would expect that $\pi(z)=0 \quad \forall z\in [0,1)$ and $\pi(1)=1$, because $\lim_{x\rightarrow \infty}z^x=0$ for $z\in [0,1)$ and $\lim_{x\rightarrow \infty}1^x=1$ with the above notion. So if we want to extend somehow the notion above, we have to add to the series something like $\mathbb{P}_X\left\{\infty \right\}$.
Is there already a common used extension for the case above?