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  "body": "\n## Definition\n\nLet $N \\ge 0$ be an integer, let $\\alpha, \\beta, \\gamma, \\delta$ be real numbers, and assume that one of $\\alpha + 1$, $\\beta + \\delta + 1$, $\\gamma + 1$ equals $-N$.\nWrite $(a)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]] and set $\\lambda(x) = x(x + \\gamma + \\delta + 1)$.\nFor $0 \\le n \\le N$, the *Racah polynomial* of degree $n$ is\n$$R_n(\\lambda(x); \\alpha, \\beta, \\gamma, \\delta) = \\sum_{k=0}^{n} \\frac{(-n)_k \\, (n + \\alpha + \\beta + 1)_k \\, (-x)_k \\, (x + \\gamma + \\delta + 1)_k}{(\\alpha + 1)_k \\, (\\beta + \\delta + 1)_k \\, (\\gamma + 1)_k \\, k!},$$\nthat is, the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_4F_3\\left( -n, n + \\alpha + \\beta + 1, -x, x + \\gamma + \\delta + 1; \\alpha + 1, \\beta + \\delta + 1, \\gamma + 1; 1 \\right)$, assuming the denominators do not vanish.\n\n## Remarks\n\nSince $(-x)_k (x + \\gamma + \\delta + 1)_k = \\prod_{j=0}^{k-1} \\left( j(j + \\gamma + \\delta + 1) - \\lambda(x) \\right)$, each $R_n$ is a polynomial of degree $n$ in $\\lambda(x)$.\nExample: $R_1 = 1 + \\frac{(\\alpha + \\beta + 2) \\, \\lambda(x)}{(\\alpha + 1)(\\beta + \\delta + 1)(\\gamma + 1)}$.\nThey are orthogonal on $x = 0, 1, \\dots, N$ for the weight\n$$w(x) = \\frac{(\\alpha + 1)_x (\\beta + \\delta + 1)_x (\\gamma + 1)_x (\\gamma + \\delta + 1)_x \\left( \\frac{\\gamma + \\delta + 3}{2} \\right)_x}{(-\\alpha + \\gamma + \\delta + 1)_x (-\\beta + \\gamma + 1)_x \\left( \\frac{\\gamma + \\delta + 1}{2} \\right)_x (\\delta + 1)_x \\, x!},$$\nthat is, $\\sum_{x=0}^{N} w(x) R_m(\\lambda(x)) R_n(\\lambda(x)) = 0$ for $m \\ne n$; the weight is positive for suitable ranges of the parameters.\nThe balanced condition holds: the sum of the lower parameters exceeds the sum of the upper ones by $1$.\nNon-example: with $\\alpha + 1 = -N$, the degree $n = N + 1$ is not allowed, since the term $k = N + 1$ has the factor $(\\alpha + 1)_{N+1} = (-N)_{N+1} = 0$ in its denominator.\n",
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