Askey–Wilson polynomials

Summary

The 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 $q$. They are defined by a terminating basic hypergeometric series ${}_4\phi_3$ in the variable $x = \cos\theta$. They sit at the top of the $q$-Askey scheme. The continuous q-Hermite polynomials, the Rogers polynomials and many other $q$-orthogonal polynomials are special or limiting cases. As $q \to 1$ they give the Wilson polynomials. Askey and Wilson introduced them in 1985.