---
tag: 25Y6SVM2D1
kind: definition
name: Wilson polynomials
msc: [33C45, 42C05]
defines: 1CPF7SWQ3S
uses: [YZHM3HSM0F, VDBAW9PN88, 0H5SYBSED9]
sources:
  - url: https://en.wikipedia.org/wiki/Wilson_polynomials
---

## Definition

Let $a, b, c, d \in \C$ and let $n \ge 0$ be an integer.
Write $(u)_k$ for the [[Pochhammer symbol|YZHM3HSM0F]].
The *Wilson polynomial* of degree $n$ is
$$W_n(x^2; a, b, c, d) = (a + b)_n (a + c)_n (a + d)_n \sum_{k=0}^{n} \frac{(-n)_k \, (n + a + b + c + d - 1)_k \, (a + ix)_k \, (a - ix)_k}{(a + b)_k \, (a + c)_k \, (a + d)_k \, k!},$$
that is, $(a + b)_n (a + c)_n (a + d)_n$ times the terminating [[generalized hypergeometric series|VDBAW9PN88]] ${}_4F_3\left( -n, n + a + b + c + d - 1, a + ix, a - ix; a + b, a + c, a + d; 1 \right)$.

## Remarks

Since $(a + ix)_k (a - ix)_k = \prod_{j=0}^{k-1} \left( (a + j)^2 + x^2 \right)$, $W_n$ is a polynomial of degree $n$ in $x^2$.
Example: $W_1(x^2) = (abc + abd + acd + bcd) - (a + b + c + d) x^2$.
Check: $(a + b)(a + c)(a + d) - (a + b + c + d) a^2 = abc + abd + acd + bcd$, and the coefficient of $x^2$ is $-(a + b + c + d)$.
This example is symmetric in $a, b, c, d$; in fact every $W_n$ is.
Orthogonality: if $a, b, c, d$ have positive real parts and the nonreal ones come in complex conjugate pairs, then for $m \ne n$
$$\int_0^\infty W_m(x^2) W_n(x^2) \abs{\frac{\Gamma(a + ix) \Gamma(b + ix) \Gamma(c + ix) \Gamma(d + ix)}{\Gamma(2ix)}}^2 dx = 0,$$
where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
The positivity hypothesis matters: if, say, $a < 0$, the integral alone no longer gives orthogonality, and the orthogonality measure needs finitely many extra point masses.
Non-example: the polynomial $x$ is not a Wilson polynomial for any parameters, since every $W_n$ is a polynomial in $x^2$ and hence an even function of $x$.