Derivatives of PGF of Discrete Uniform Distribution

Theorem
Let $X$ be a discrete random variable with the discrete uniform distribution with parameter $n$.

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


 * $\displaystyle \frac {d^m} {ds^m} \Pi_X \left({s}\right) = \begin{cases}

\displaystyle \frac 1 n \sum_{k=m}^n k^{\underline m} s^{k-m} & : m \le n \\ 0 & : k > n \end{cases}$ where $k^{\underline m}$ is the falling factorial.

Proof
The Probability Generating Function of Discrete Uniform Distribution is:
 * $\displaystyle \Pi_X \left({s}\right) = \frac {s \left({1 - s^n}\right)} {n \left({1 - s}\right)} = \frac 1 n \sum_{k=1}^n s^k$

From Nth Derivative of Mth Power:
 * $\displaystyle \frac {d^k} {ds^k} s^n = \begin{cases}

n^{\underline k} s^{n-k} & : k \le n \\ 0 & : k > n \end{cases}$

The result follows.