Derivatives of Probability Generating Function at One

Theorem
Let $X$ be a discrete random variable whose probability generating function is $\Pi_X \left({s}\right)$.

Then the $n$th derivative of $\Pi_X \left({s}\right)$ at $s = 1$ is given by:
 * $\dfrac {\mathrm d^n} {\mathrm d s^n} \Pi_X \left({1}\right) = E \left({X \left({X - 1}\right) \cdots \left({X - n + 1}\right)}\right)$

for $n = 1, 2, \ldots$

Proof
Proof by induction:

For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * $\dfrac {\mathrm d^n} {\mathrm d s^n} \Pi_X \left({1}\right) = E \left({X \left({X - 1}\right) \cdots \left({X - n + 1}\right)}\right)$

Basis for the Induction
$P \left({1}\right)$ is the case:
 * $\dfrac {\mathrm d} {\mathrm d s} \Pi_X \left({1}\right) = E \left({X}\right)$

for $n = 1, 2, \ldots$

This is demonstrated in Expectation of Discrete Random Variable from PGF.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 2$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({1}\right) = E \left({X \left({X - 1}\right) \cdots \left({X - k + 1}\right)}\right)$

Then we need to show:
 * $\dfrac {\mathrm d^{k+1}} {\mathrm d s^{k+1}} \Pi_X \left({1}\right) = E \left({X \left({X - 1}\right) \cdots \left({X - k}\right)}\right)$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $proposition_n$