Hahn polynomial

The Hahn polynomials are defined by $$Q_n(x;\alpha,\beta,N)={}_3F_2 (-n,n+\alpha+\beta+1,-x;\alpha+1,-N;1),$$ where ${}_3F_2$ denotes the generalized hypergeometric function.

Properties

Theorem: (Orthogonality) The Hahn polynomials are orthogonal with respect to the inner product $$\langle Q_m,Q_n \rangle = \displaystyle\sum_{k=0}^N Q_m(k;\alpha,\beta,N)Q_n(k;\alpha,\beta,N)\dfrac{(\alpha+1)_k}{k!} \dfrac{(\beta+1)_{N-k}}{(N-k)!},$$ and obey the formula $$\langle Q_m,Q_n \rangle=\dfrac{n!(N-n)!(\beta+1)_n(\alpha+\beta+n+1)_{N+1}}{(N!)^2(\alpha+\beta+2n+1)(\alpha+1)_n}\delta_{mn}.$$

Proof: █