Second Derivative of PGF of Negative Binomial Distribution/First Form

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

Then the second derivative of the PGF of $X$ w.r.t. $s$ is:


 * $\dfrac {\mathrm d^2} {\mathrm d s^2} \Pi_X \left({s}\right) = \dfrac {n \left({n + 1}\right) p^2} {q^2} \left({\dfrac q {1 - ps} }\right)^{n+2}$

where $q = 1 - p$.

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


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

From Derivatives of PGF of Negative Binomial Distribution/First Form:
 * $(1): \quad \dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \dfrac {n^{\overline k} p^k} {q^k} \left({\dfrac q {1 - ps} }\right)^{n+k}$

where:
 * $n^{\overline k}$ is the rising factorial: $n^{\overline k} = n \left({n+1}\right) \left({n+2}\right) \cdots \left({n+k-1}\right)$
 * $q = 1 - p$.

Putting $k = 2$ in $(1)$ above yields the required solution.