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  "body": "\n## Definition\n\nLet $N \\ge 0$ be an integer, let $\\alpha, \\beta \u003e -1$ be real numbers, and let $n$ be an integer with $0 \\le n \\le N$.\nWrite $(a)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].\nThe *Hahn polynomial* of degree $n$ is\n$$Q_n(x; \\alpha, \\beta, N) = \\sum_{k=0}^{n} \\frac{(-n)_k \\, (n + \\alpha + \\beta + 1)_k \\, (-x)_k}{(\\alpha + 1)_k \\, (-N)_k \\, k!},$$\nthat is, the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_3F_2\\left( -n, n + \\alpha + \\beta + 1, -x; \\alpha + 1, -N; 1 \\right)$.\n\n## Remarks\n\nThey are orthogonal on $\\set{0, 1, \\dots, N}$:\n$$\\sum_{x=0}^{N} \\binom{\\alpha + x}{x} \\binom{\\beta + N - x}{N - x} Q_m(x) Q_n(x) = 0 \\quad (m \\ne n),$$\nwhere the [[binomial coefficients|0GBZM0MT0F]] are $\\binom{a + j}{j} = \\frac{(a + 1)_j}{j!}$.\nExample: $Q_1(x) = 1 - \\frac{(\\alpha + \\beta + 2) x}{(\\alpha + 1) N}$.\nCheck: the weight is proportional to a beta-binomial distribution with mean $\\frac{N(\\alpha + 1)}{\\alpha + \\beta + 2}$, so $Q_1$ has weighted mean $0$.\nNon-example: $n = N + 1$ is not allowed, since $(-N)_k = 0$ for $k = N + 1$ and the defining sum has a zero denominator.\nThe value at $0$ is $Q_n(0) = 1$.\n",
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    "name": "Hahn polynomials",
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  "name": "Hahn polynomials",
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