---
tag: 1CPF7SWQ3S
kind: concept
name: Wilson polynomials
msc: [33C45, 42C05]
sources:
  - url: https://en.wikipedia.org/wiki/Wilson_polynomials
same_as:
  wikidata: Q8023263
  wikipedia: 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|0H5SYBSED9]].
They are at the top of the [[Askey scheme|RR2S05C09B]].
Every other family in the scheme arises from them, or from the [[Racah polynomials|P6JYNDX5DR]], by specialisation or limits.
Their $q$-analogues are the [[Askey–Wilson polynomials|RSNZNG0WA1]].

## Explanations

### L1 human

The classical orthogonal polynomials, such as those of Jacobi, Laguerre and Hermite, come in a few families with few parameters.
The Wilson polynomials are one family with four parameters from which all the others can be reached.
Setting a parameter to a special value, or letting it go to infinity, moves down from the Wilson polynomials to the more familiar ones.
They are symmetric in their four parameters, which is not visible from the formula but is a deep identity.