Theorem 6.5.3. Moment Generating Function for Hypergeometric.
For a hypergeometric random variable over the space \(R\) = {0, 1, ..., min(\(r,n_1\))}.
\begin{equation*}
M(t) = \sum ( e^{tx} \frac{\binom{n_1}{x} \binom{n-n_1}{r-x}}{\binom{n}{r}}).
\end{equation*}
There is not a single easy formula that captures this summation nicely but with particular values for the parameters perhaps something could be simplified.