Derivatives of PGF of Negative Binomial Distribution/Second Form

Theorem
Let $X$ be a discrete random variable with the negative binomial distribution (second form) with parameters $n$ and $p$.

Then the derivatives of the PGF of $X$ w.r.t. $s$ are:


 * $\dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = ...$

Proof
The Probability Generating Function of Negative Binomial Distribution (Second Form) is:


 * $\Pi_X \left({s}\right) = \left({\dfrac {p s} {1 - q s}}\right)^n$

We have that for a given negative binomial distribution, $n, p$ and $q$ are constant.