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  "body": "\n## Summary\n\nThe Hahn polynomials $Q_n(x; \\alpha, \\beta, N)$ are a finite family of orthogonal polynomials on the points $0, 1, \\dots, N$.\nTheir weight is the beta-binomial distribution, a discrete version of the Jacobi weight.\nThey are terminating ${}_3F_2$ hypergeometric series and sit in the [[Askey scheme|RR2S05C09B]] just below the Wilson and Racah polynomials.\nAs $N \\to \\infty$ after rescaling they tend to the [[Jacobi polynomials|TRD7DSF2KR]], and they also appear as Clebsch–Gordan coefficients of $SU(2)$.\n\n## Explanations\n\n### L1 human\n\nThe Jacobi polynomials are perpendicular for a weight on a continuous interval.\nReplace the interval by the $N + 1$ points $0, 1, \\dots, N$ and the weight by a matching discrete weight.\nThe polynomials that are perpendicular for this discrete weight are the Hahn polynomials.\nFor each degree $n$ from $0$ to $N$ there is exactly one of them, normalised to take the value $1$ at $x = 0$.\n",
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