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Krawtchouk Polynomials
Definition
The Krawtchouk polynomials are defined as
\[
\begin{align}
K_n(x;p,N) &= {N \choose n}^{-1} \sum_{k=0}^n {N-x \choose n-k} {x \choose k} \left(1-\frac{1}{p}\right)^k\\
&= {}_2F_1 \left(\left. {-n, -x \atop -N} \; \right| \frac{1}{p} \right)
\end{align}
\]
