Probability Generating Function of Zero

Theorem
Let $X$ be a discrete random variable whose codomain, $\Omega_X$, is a subset of the natural numbers $\N$.

Let $p_X$ be the probability mass function for $X$.

Let $\map {\Pi_X} s$ be the probability generating function for $X$.

Then:
 * $\map {\Pi_X} 0 = \map {p_X} 0$