Hahn polynomials

Remarks

They are orthogonal on $\set{0, 1, \dots, N}$: $$\sum_{x=0}^{N} \binom{\alpha + x}{x} \binom{\beta + N - x}{N - x} Q_m(x) Q_n(x) = 0 \quad (m \ne n),$$ where the binomial coefficients are $\binom{a + j}{j} = \frac{(a + 1)_j}{j!}$. Example: $Q_1(x) = 1 - \frac{(\alpha + \beta + 2) x}{(\alpha + 1) N}$. Check: the weight is proportional to a beta-binomial distribution with mean $\frac{N(\alpha + 1)}{\alpha + \beta + 2}$, so $Q_1$ has weighted mean $0$. Non-example: $n = N + 1$ is not allowed, since $(-N)_k = 0$ for $k = N + 1$ and the defining sum has a zero denominator. The value at $0$ is $Q_n(0) = 1$.