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  "body": "\n## Definition\n\nLet $q \\in \\C$ with $0 \u003c \\abs{q} \u003c 1$, and let $a, b, c, d \\in \\C$ with $a \\ne 0$.\nWrite $(c_1, \\dots, c_k; q)_n = (c_1; q)_n \\cdots (c_k; q)_n$ with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\nFor an integer $n \\ge 0$ and $x = \\cos\\theta$, the *Askey–Wilson polynomial* of degree $n$ is\n$$p_n(x; a, b, c, d \\mid q) = a^{-n} (ab, ac, ad; q)_n\\; {}_4\\phi_3 \\left( \\begin{matrix} q^{-n},\\ abcd\\, q^{n-1},\\ a e^{i\\theta},\\ a e^{-i\\theta} \\\\ ab,\\ ac,\\ ad \\end{matrix} ; q, q \\right),$$\nwhere ${}_4\\phi_3$ is the terminating [[basic hypergeometric series|EB4N9RQR51]].\n\n## Remarks\n\nSince $(a e^{i\\theta}, a e^{-i\\theta}; q)_k = \\prod_{j=0}^{k-1} (1 - 2 a q^j x + a^2 q^{2j})$, each term is a polynomial in $x$, and $p_n$ is a polynomial of degree at most $n$ with leading coefficient $2^n (abcd\\, q^{n-1}; q)_n$.\nThe factor $a^{-n} (ab, ac, ad; q)_n$ cancels the denominators, and $p_n$ is symmetric in $a, b, c, d$.\n\nExample: $n = 1$.\nThe ${}_4\\phi_3$ has two terms, and since $\\frac{(1 - q^{-1})\\, q}{1 - q} = -1$,\n$$p_1(x) = a^{-1} \\left[ (1 - ab)(1 - ac)(1 - ad) - (1 - abcd)(1 - 2ax + a^2) \\right] = 2(1 - abcd)\\, x + (abc + abd + acd + bcd) - (a + b + c + d).$$\n\nOrthogonality: if $0 \u003c q \u003c 1$ and $a, b, c, d$ are real with absolute values less than $1$, then for $m \\ne n$\n$$\\int_{-1}^{1} p_m(x)\\, p_n(x)\\, w(x)\\, \\frac{dx}{\\sqrt{1 - x^2}} = 0, \\qquad w(\\cos\\theta) = \\frac{(e^{2i\\theta}, e^{-2i\\theta}; q)_\\infty}{(a e^{i\\theta}, a e^{-i\\theta}, b e^{i\\theta}, b e^{-i\\theta}, c e^{i\\theta}, c e^{-i\\theta}, d e^{i\\theta}, d e^{-i\\theta}; q)_\\infty}.$$\n\nNon-example: for parameters as in the orthogonality statement, $x^2$ is not $p_2$, since $p_2$ is orthogonal to the constant $p_0 = 1$ for the positive weight $w$, while $\\int x^2 w(x) \\frac{dx}{\\sqrt{1-x^2}} \u003e 0$.\n",
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    "name": "Askey–Wilson polynomials",
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