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  "body": "\n## Summary\n\nThe Askey–Wilson polynomials $p_n(x; a, b, c, d \\mid q)$ are a family of orthogonal polynomials in one variable with four parameters and a base $q$.\nThey are defined by a terminating [[basic hypergeometric series|EB4N9RQR51]] ${}_4\\phi_3$ in the variable $x = \\cos\\theta$.\nThey sit at the top of the $q$-Askey scheme.\nThe [[continuous q-Hermite polynomials|XFKVZJ55Q5]], the [[Rogers polynomials|7H7SACRYEF]] and many other $q$-orthogonal polynomials are special or limiting cases.\nAs $q \\to 1$ they give the [[Wilson polynomials|1CPF7SWQ3S]].\nAskey and Wilson introduced them in 1985.\n\n## Explanations\n\n### L1 human\n\nClassical orthogonal polynomials, like the Legendre or Chebyshev polynomials, are each defined by a formula with few parameters.\nThe Askey–Wilson polynomials are one formula with five parameters $a, b, c, d, q$ that contains a large number of them as special cases or limits.\nThey are orthogonal on $[-1, 1]$ for a weight built from infinite products.\nFor example, the polynomial of degree $1$ is $2(1 - abcd)\\, x + (abc + abd + acd + bcd) - (a + b + c + d)$, which is symmetric in the four parameters.\n",
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