Derivatives of PGF of Negative Binomial Distribution/First 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 $X$ $s$ are:


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

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

Proof
Proof by induction:

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


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

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

Basis for the Induction
$P \left({1}\right)$ is the case:
 * $\dfrac {\mathrm d} {\mathrm d s} \Pi_X \left({s}\right) = \dfrac {n^{\overline 1} p} q \left({\dfrac q {1 - p s} }\right)^{n + 1}$

which is proved in First Derivative of PGF of Negative Binomial Distribution: First Form.

Note that from Number to Power of One Rising is Itself:
 * $n^\overline 1 = n$

This is our basis for the induction.

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

So this is our induction hypothesis:
 * $\dfrac {\mathrm d^j} {\mathrm d s^j} \Pi_X \left({s}\right) = \dfrac {n^{\overline j} p^j} {q^j} \left({\dfrac q {1 - p s} }\right)^{n + j}$

Then we need to show:
 * $\dfrac {\mathrm d^{j + 1} } {\mathrm d s^{j + 1} } \Pi_X \left({s}\right) = \dfrac {n^{\overline {j + 1}} p^{j + 1} } {q^{j + 1} } \left({\dfrac q {1 - p s} }\right)^{n + j + 1}$

Induction Step
This is our induction step:

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

Therefore:
 * $\forall k \in \N_{> 0}: \dfrac {\mathrm d^k} {\mathrm d s^k} \Pi_X \left({s}\right) = \dfrac {n^{\overline k} p^k} {q^k} \left({\dfrac q {1 - p s} }\right)^{n + k}$