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  "body": "\n## Definition\n\nLet $a, b, c, d \\in \\C$ and let $n \\ge 0$ be an integer.\nWrite $(u)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].\nThe *Wilson polynomial* of degree $n$ is\n$$W_n(x^2; a, b, c, d) = (a + b)_n (a + c)_n (a + d)_n \\sum_{k=0}^{n} \\frac{(-n)_k \\, (n + a + b + c + d - 1)_k \\, (a + ix)_k \\, (a - ix)_k}{(a + b)_k \\, (a + c)_k \\, (a + d)_k \\, k!},$$\nthat is, $(a + b)_n (a + c)_n (a + d)_n$ times the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_4F_3\\left( -n, n + a + b + c + d - 1, a + ix, a - ix; a + b, a + c, a + d; 1 \\right)$.\n\n## Remarks\n\nSince $(a + ix)_k (a - ix)_k = \\prod_{j=0}^{k-1} \\left( (a + j)^2 + x^2 \\right)$, $W_n$ is a polynomial of degree $n$ in $x^2$.\nExample: $W_1(x^2) = (abc + abd + acd + bcd) - (a + b + c + d) x^2$.\nCheck: $(a + b)(a + c)(a + d) - (a + b + c + d) a^2 = abc + abd + acd + bcd$, and the coefficient of $x^2$ is $-(a + b + c + d)$.\nThis example is symmetric in $a, b, c, d$; in fact every $W_n$ is.\nOrthogonality: if $a, b, c, d$ have positive real parts and the nonreal ones come in complex conjugate pairs, then for $m \\ne n$\n$$\\int_0^\\infty W_m(x^2) W_n(x^2) \\abs{\\frac{\\Gamma(a + ix) \\Gamma(b + ix) \\Gamma(c + ix) \\Gamma(d + ix)}{\\Gamma(2ix)}}^2 dx = 0,$$\nwhere $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nThe positivity hypothesis matters: if, say, $a \u003c 0$, the integral alone no longer gives orthogonality, and the orthogonality measure needs finitely many extra point masses.\nNon-example: the polynomial $x$ is not a Wilson polynomial for any parameters, since every $W_n$ is a polynomial in $x^2$ and hence an even function of $x$.\n",
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    "name": "Wilson polynomials",
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