Krawtchouk polynomials

Remarks

The denominators $(-N)_k$ are nonzero because $k \le n \le N$. They are orthogonal for the binomial distribution: $$\sum_{x=0}^{N} \binom{N}{x} p^x (1 - p)^{N - x} K_m(x; p, N) K_n(x; p, N) = \frac{\delta_{mn}}{\binom{N}{n}} \left( \frac{1 - p}{p} \right)^n.$$ Example: $K_1(x; p, N) = 1 - \frac{x}{Np}$. Check: if $X$ is binomial with mean $Np$ and variance $Np(1 - p)$, then $E[K_1(X)] = 0$ and $E[K_1(X)^2] = \frac{Np(1 - p)}{N^2 p^2} = \frac{1}{N} \cdot \frac{1 - p}{p}$. Non-example: the clause $n \le N$ matters; for $n > N$ the formula breaks down at $k = N + 1$, where $(-N)_k = 0$. In coding theory one often uses the version $\mathcal{K}_k(x) = \sum_{j=0}^{k} (-1)^j (q - 1)^{k - j} \binom{x}{j} \binom{N - x}{k - j}$ for an alphabet of size $q$, which is a constant multiple of $K_k(x; 1 - 1/q, N)$.