CAOP

$$\begin{align} K_n(x;p,N) &= {N \choose n}^{-1} \sum_{k=0}^n {N-x \choose n-k} {x \choose k} \left(1-\frac{1}{p}\right)^k\\ &= {}_2F_1 \left(\left. {-n, -x \atop -N} \; \right| \frac{1}{p} \right) \end{align}$$