# Wilson

## Hypergeometric representation

$\displaystyle {\displaystyle \frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}} {}=\HyperpFq{4}{3}@@{-n,n+a+b+c+d-1,a+\iunit x,a-\iunit x}{a+b,a+c,a+d}{1} }$

## Orthogonality relation(s)

$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_0^{\infty} \left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}\EulerGamma@{d+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2 {} \Wilson{m}@{x^2}{a}{b}{c}{d}\Wilson{n}@{x^2}{a}{b}{c}{d}\,dx {}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n} }$
$\displaystyle {\displaystyle \EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d} {}=\EulerGamma@{n+a+b}\EulerGamma@{n+a+c}\EulerGamma@{n+a+d}\EulerGamma@{n+b+c}\EulerGamma@{n+b+d}\EulerGamma@{n+c+d} }$
$\displaystyle {\displaystyle \frac{1}{2\cpi}\int_0^{\infty}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}\EulerGamma@{d+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2 {} \Wilson{m}@{x^2}{a}{b}{c}{d}\Wilson{n}@{x^2}{a}{b}{c}{d}\,dx {}+\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b-a}\EulerGamma@{c-a}\EulerGamma@{d-a}}{\EulerGamma@{-2a}} {}\sum_{\begin{array}{c} {\scriptstyle k=0,1,2\ldots}\\ {\scriptstyle a+k<0}\end{array}} \frac{\pochhammer{2a}{k}\pochhammer{a+1}{k}\pochhammer{a+b}{k}\pochhammer{a+c}{k}\pochhammer{a+d}{k}}{\pochhammer{a}{k}\pochhammer{a-b+1}{k}\pochhammer{a-c+1}{k}\pochhammer{a-d+1}{k}k!} {} \Wilson{m}@{-(a+k)^2}{a}{b}{c}{d}\Wilson{n}@{-(a+k)^2}{a}{b}{c}{d} {}=\frac{\EulerGamma@{n+a+b}\cdots\EulerGamma@{n+c+d}}{\EulerGamma@{2n+a+b+c+d}}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n} }$

## Recurrence relation

$\displaystyle {\displaystyle -\left(a^2+x^2\right)\normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d} =A_n\normWilsonWtilde{n+1}@@{x^2}{a}{b}{c}{d}-\left(A_n+C_n\right)\normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d}+C_n\normWilsonWtilde{n-1}@@{x^2}{a}{b}{c}{d} }$

Substitution(s): $\displaystyle {\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}$ &
$\displaystyle {\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}$

$\displaystyle {\displaystyle \normWilsonWtilde{n}@@{x^2}{a}{b}{c}{d}:=\normWilsonWtilde{n}@{x^2}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}} }$

## Monic recurrence relation

$\displaystyle {\displaystyle x\monicWilson{n}@@{x}{a}{b}{c}{d}=\monicWilson{n+1}@@{x}{a}{b}{c}{d}+(A_n+C_n-a^2)\monicWilson{n}@@{x}{a}{b}{c}{d}+A_{n-1}C_n\monicWilson{n-1}@@{x}{a}{b}{c}{d} }$

Substitution(s): $\displaystyle {\displaystyle C_n=\frac{n(n+b+c-1)(n+b+d-1)(n+c+d-1)}{(2n+a+b+c+d-2)(2n+a+b+c+d-1)}}$ &
$\displaystyle {\displaystyle A_n=\frac{(n+a+b+c+d-1)(n+a+b)(n+a+c)(n+a+d)}{(2n+a+b+c+d-1)(2n+a+b+c+d)}}$

$\displaystyle {\displaystyle \Wilson{n}@{x^2}{a}{b}{c}{d}=(-1)^n\pochhammer{n+a+b+c+d-1}{n}\monicWilson{n}@@{x^2}{a}{b}{c}{d} }$

## Difference equation

$\displaystyle {\displaystyle n(n+a+b+c+d-1)y(x) {}=B(x)y(x+\iunit)-\left[B(x)+D(x)\right]y(x)+D(x)y(x-\iunit) }$

Substitution(s): $\displaystyle {\displaystyle D(x)=\frac{(a+\iunit x)(b+\iunit x)(c+\iunit x)(d+\iunit x)}{2\iunit x(2\iunit x+1)}}$ &

$\displaystyle {\displaystyle B(x)=\frac{(a-\iunit x)(b-\iunit x)(c-\iunit x)(d-\iunit x)}{2\iunit x(2\iunit x-1)}}$ &

$\displaystyle {\displaystyle y(x)=\Wilson{n}@{x^2}{a}{b}{c}{d}}$

## Forward shift operator

$\displaystyle {\displaystyle \Wilson{n}@{(x+\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c}{d} -\Wilson{n}@{(x-\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c}{d} {}=-2\iunit nx(n+a+b+c+d-1)\Wilson{n-1}@{x^2}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}} }$
$\displaystyle {\displaystyle \frac{\delta \Wilson{n}@{x^2}{a}{b}{c}{d}}{\delta x^2} {}=-n(n+a+b+c+d-1)\Wilson{n-1}@{x^2}{a+\textstyle\frac{1}{2}}{b+\textstyle\frac{1}{2}}{c+\textstyle\frac{1}{2}}{d+\textstyle\frac{1}{2}} }$

## Backward shift operator

$\displaystyle {\displaystyle (a-\textstyle\frac{1}{2}-\iunit x)(b-\textstyle\frac{1}{2}-\iunit x) (c-\textstyle\frac{1}{2}-\iunit x)(d-\textstyle\frac{1}{2}-\iunit x) \Wilson{n}@{(x+\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c}{d} {}-(a-\textstyle\frac{1}{2}+\iunit x)(b-\textstyle\frac{1}{2}+\iunit x) (c-\textstyle\frac{1}{2}+\iunit x)(d-\textstyle\frac{1}{2}+\iunit x) \Wilson{n}@{(x-\textstyle\frac{1}{2}\iunit)^2}{a}{b}{c}{d} {}=-2\iunit x \Wilson{n+1}@{x^2}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}} }$
$\displaystyle {\displaystyle \frac{\delta\left[\omega(x;a,b,c,d)\Wilson{n}@{x^2}{a}{b}{c}{d}\right]}{\delta x^2} {}=\omega(x;a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2} c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}) \Wilson{n+1}@{x^2}{a-\textstyle\frac{1}{2}}{b-\textstyle\frac{1}{2}}{c-\textstyle\frac{1}{2}}{d-\textstyle\frac{1}{2}} }$

Substitution(s): $\displaystyle {\displaystyle \omega(x;a,b,c,d):=\frac{1}{2\iunit x}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}\EulerGamma@{d+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2}$

## Rodrigues-type formula

$\displaystyle {\displaystyle \omega(x;a,b,c,d)\Wilson{n}@{x^2}{a}{b}{c}{d} {}=\left(\frac{\delta}{\delta x^2}\right)^n \left[\omega(x;a+\textstyle\frac{1}{2}n,b+\textstyle\frac{1}{2}n c+\textstyle\frac{1}{2}n,d+\textstyle\frac{1}{2}n)\right] }$

Substitution(s): $\displaystyle {\displaystyle \omega(x;a,b,c,d):=\frac{1}{2\iunit x}\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}\EulerGamma@{d+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2}$

## Generating functions

$\displaystyle {\displaystyle \HyperpFq{2}{1}@@{a+\iunit x,b+\iunit x}{a+b}{t}\,\HyperpFq{2}{1}@@{c-\iunit x,d-\iunit x}{c+d}{t}= \sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}t^n}{\pochhammer{a+b}{n}\pochhammer{c+d}{n}n!} }$
$\displaystyle {\displaystyle \HyperpFq{2}{1}@@{a+\iunit x,c+\iunit x}{a+c}{t}\,\HyperpFq{2}{1}@@{b-\iunit x,d-\iunit x}{b+d}{t}= \sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}t^n}{\pochhammer{a+c}{n}\pochhammer{b+d}{n}n!} }$
$\displaystyle {\displaystyle \HyperpFq{2}{1}@@{a+\iunit x,d+\iunit x}{a+d}{t}\,\HyperpFq{2}{1}@@{b-\iunit x,c-\iunit x}{b+c}{t}= \sum_{n=0}^{\infty}\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}t^n}{\pochhammer{a+d}{n}\pochhammer{b+c}{n}n!} }$
$\displaystyle {\displaystyle (1-t)^{1-a-b-c-d} {}\HyperpFq{4}{3}@@{\frac{1}{2}(a+b+c+d-1),\frac{1}{2}(a+b+c+d),a+\iunit x,a-\iunit x}{a+b,a+c,a+d}{-\frac{4t}{(1-t)^2}} {}=\sum_{n=0}^{\infty}\frac{\pochhammer{a+b+c+d-1}{n}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}n!}\Wilson{n}@{x^2}{a}{b}{c}{d}t^n }$

## Limit relations

### Wilson polynomial to Continuous dual Hahn polynomial

$\displaystyle {\displaystyle \lim_{d\rightarrow\infty}\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+d}{n}}=\ctsdualHahn{n}@{x^2}{a}{b}{c} }$

### Wilson polynomial to Continuous Hahn polynomial

$\displaystyle {\displaystyle \lim_{t\rightarrow\infty} \frac{\Wilson{n}@{(x+t)^2}{a-\iunit t}{b-\iunit t}{c+\iunit t}{d+\iunit t}}{(-2t)^nn!}=\ctsHahn{n}@{x}{a}{b}{c}{d} }$

### Wilson polynomial to Jacobi polynomial

$\displaystyle {\displaystyle \lim_{t\rightarrow\infty}\frac{\Wilson{n}@{\frac{1}{2}(1-x)t^2}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\alpha+1)}{\frac{1}{2}(\beta+1)+\iunit t}{\frac{1}{2}(\beta+1}-\iunit t)}{t^{2n}n!} {}=\Jacobi{\alpha}{\beta}{n}@{x} }$

## Remarks

$\displaystyle {\displaystyle \frac{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{\pochhammer{a+b}{k}\pochhammer{a+c}{k}\pochhammer{a+d}{k}}=\pochhammer{a+b+k}{n-k}\pochhammer{a+c+k}{n-k}\pochhammer{a+d+k}{n-k} }$
$\displaystyle {\displaystyle a=\textstyle\frac{1}{2}(\gamma+\delta+1) }$
$\displaystyle {\displaystyle b=\textstyle\frac{1}{2}(2\alpha-\gamma-\delta+1) }$
$\displaystyle {\displaystyle c=\textstyle\frac{1}{2}(2\beta-\gamma+\delta+1) }$
$\displaystyle {\displaystyle d=\textstyle\frac{1}{2}(\gamma-\delta+1) }$
$\displaystyle {\displaystyle \iunit x\rightarrow x+\textstyle\frac{1}{2}(\gamma+\delta+1) }$
$\displaystyle {\displaystyle \normWilsonWtilde{n}@{x^2}{a}{b}{c}{d}=\frac{\Wilson{n}@{x^2}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}} }$
$\displaystyle {\displaystyle \alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N }$

$\displaystyle {\displaystyle \Wilson{n}@{x^2}{a}{b}{c}{d} =\frac{\pochhammer{a-\iunit x}{n} \pochhammer{b-\iunit x}{n} \pochhammer{c-\iunit x}{n} \pochhammer{d-\iunit x}{n}}{\pochhammer{-2\iunit x}{n}} \HyperpFq{7}{6}@@{2\iunit x-n,\iunit x-\frac12 n+1,a+\iunit x,b+\iunit x,c+\iunit x,d+\iunit x,-n}{\iunit x-\frac12 n,1-n-a+\iunit x,1-n-b+\iunit x,1-n-c+\iunit x,1-n-d+\iunit x}{1} }$
$\displaystyle {\displaystyle \Wilson{n}@{-a^2}{a}{b}{c}{d}=\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n} }$
$\displaystyle {\displaystyle \frac{|\Wilson{n}@{-a^2}{a}{b}{c}{d}|^2}{h_n} = O(n^{4\realpart{a}-1}) \hbox{as n\to\infty.} }$