= Solution
The <probability generating function> of a nonnegative <integer>-valued <random variable> is
$$
\boxed{G_X(s)=\mathbb E[s^X]=\sum_{k=0}^{\infty}p_X(k)s^k,}
$$
using $s^0=1$ also at $s=0$. It is always defined for $0\leq s\leq1$, and indeed the <power series> converges absolutely for all real or complex $s$ with $|s|\leq1$, because
$$
\sum_{k=0}^{\infty}p_X(k)|s|^k\leq\sum_{k=0}^{\infty}p_X(k)=1.
$$
It may be defined beyond the unit disk where this <power series> converges absolutely; such a larger <domain> depends on the <distribution>. In particular $G_X(0)=\mathbb P(X=0)$ and $G_X(1)=1$.
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