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:


 * $$\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:
 * $$\Pi_X \left({s}\right) = \frac {s \left({1 - s^n}\right)} {n \left({1 - s}\right)} = \frac 1 n \sum_{k=1}^n s^k$$

From Nth Derivative of Mth Power:
 * $$\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.