Wilson polynomials

Summary

The Wilson polynomials $W_n(x^2; a, b, c, d)$ are orthogonal polynomials in the variable $x^2$ with four parameters, introduced by James Wilson in 1980. They are given by terminating balanced ${}_4F_3$ hypergeometric series, and their weight on $(0, \infty)$ is a ratio of products of gamma functions. They are at the top of the Askey scheme. Every other family in the scheme arises from them, or from the Racah polynomials, by specialisation or limits. Their $q$-analogues are the Askey–Wilson polynomials.