Derivatives of PGF of Binomial Distribution

Theorem
Let $$X$$ be a discrete random variable with the binomial distribution with parameters $n$ and $p$.

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


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

n^{\underline k} p^k \left({q + ps}\right)^{n-k} & : k \le n \\ 0 & : k > n \end{cases}$$ where:
 * $$n^{\underline k}$$ is the falling factorial;
 * $$q = 1 - p$$.

Proof
The Probability Generating Function of Binomial Distribution is:
 * $$\Pi_X \left({s}\right) = \left({q + ps}\right)^n$$

where $$q = 1 - p$$.

From Derivatives of Function of ax + b, we have that:
 * $$\frac {d^k} {ds^k} \left({f \left({q + ps}\right)}\right) = p^k \frac {d^k} {dz^k} \left({f \left({z}\right)}\right)$$

where $$z = q + ps$$.

Here we have that $$f \left({z}\right) = z^n$$.

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

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

So putting it together:
 * $$\frac {d^k} {ds^k} \Pi_X \left({s}\right) = \begin{cases}

n^{\underline k} p^k \left({q + ps}\right)^{n-k} & : k \le n \\ 0 & : k > n \end{cases}$$