Wilson

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Wilson

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Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\frac{\Wilson{n}@{x^{2}}{a}{b}{c}{d}}{\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}{}={{}_{4}F_{3}}\!\left({-n,n+a+b+c+d-1,a+\iunit x,a-\iunit x\atop a+b,a+c,a+d};1\right)}}}$

Orthogonality relation(s)

${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\cpi}\int_{0}^{\infty}\left|\frac{\Gamma\left(a+\iunit x\right)\Gamma\left(b+\iunit x\right)\Gamma\left(c+\iunit x\right)\Gamma\left(d+\iunit x\right)}{\Gamma\left(2\iunit x\right)}\right|^{2}{}\Wilson{m}@{x^{2}}{a}{b}{c}{d}\Wilson{n}@{x^{2}}{a}{b}{c}{d}\,dx{}=\frac{\Gamma\left(n+a+b\right)\cdots\Gamma\left(n+c+d\right)}{\Gamma\left(2n+a+b+c+d\right)}\pochhammer{n+a+b+c+d-1}{n}n!\,\Kronecker{m}{n}}}}$
${\displaystyle{\displaystyle{\displaystyle\Gamma\left(n+a+b\right)\cdots\Gamma\left(n+c+d\right){}=\Gamma\left(n+a+b\right)\Gamma\left(n+a+c\right)\Gamma\left(n+a+d\right)\Gamma\left(n+b+c\right)\Gamma\left(n+b+d\right)\Gamma\left(n+c+d\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}\left|\frac{\Gamma\left(a+\mathrm{i}x\right)\Gamma\left(b+\mathrm{i}x\right)\Gamma\left(c+\mathrm{i}x\right)\Gamma\left(d+\mathrm{i}x\right)}{\Gamma\left(2\mathrm{i}x\right)}\right|^{2}{}W_{m}\!\left(x^{2};a,b,c,d\right)W_{n}\!\left(x^{2};a,b,c,d\right)\,dx{}+\frac{\Gamma\left(a+b\right)\Gamma\left(a+c\right)\Gamma\left(a+d\right)\Gamma\left(b-a\right)\Gamma\left(c-a\right)\Gamma\left(d-a\right)}{\Gamma\left(-2a\right)}{}\sum_{\begin{array}[]{c}{\scriptstyle k=0,1,2\ldots}\\ {\scriptstyle a+k<0}\end{array}}\frac{{\left(2a\right)_{k}}{\left(a+1\right)_{k}}{\left(a+b\right)_{k}}{\left(a+c\right)_{k}}{\left(a+d\right)_{k}}}{{\left(a\right)_{k}}{\left(a-b+1\right)_{k}}{\left(a-c+1\right)_{k}}{\left(a-d+1\right)_{k}}k!}{}W_{m}\!\left(-(a+k)^{2};a,b,c,d\right)W_{n}\!\left(-(a+k)^{2};a,b,c,d\right){}=\frac{\Gamma\left(n+a+b\right)\cdots\Gamma\left(n+c+d\right)}{\Gamma\left(2n+a+b+c+d\right)}{\left(n+a+b+c+d-1\right)_{n}}n!\,\delta_{m,n}}}}$

Recurrence relation

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

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

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

Monic recurrence relation

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

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

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

Difference equation

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

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

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

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

Forward shift operator

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

Backward shift operator

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

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

Rodrigues-type formula

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

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

Generating functions

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

Limit relations

Wilson polynomial to Continuous dual Hahn polynomial

${\displaystyle{\displaystyle{\displaystyle\lim_{d\rightarrow\infty}\frac{W_{n}\!\left(x^{2};a,b,c,d\right)}{{\left(a+d\right)_{n}}}=S_{n}\!\left(x^{2};a,b,c\right)}}}$

Wilson polynomial to Continuous Hahn polynomial

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

Wilson polynomial to Jacobi polynomial

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

Remarks

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

Koornwinder Addendum: Wilson

Hypergeometric representation

${\displaystyle{\displaystyle{\displaystyle\Wilson{n}@{x^{2}}{a}{b}{c}{d}=\frac{\pochhammer{a-\iunit x}{n}\pochhammer{b-\iunit x}{n}\pochhammer{c-\iunit x}{n}\pochhammer{d-\iunit x}{n}}{\pochhammer{-2\iunit x}{n}}{{}_{7}F_{6}}\!\left({2\iunit x-n,\iunit x-\frac{1}{2}n+1,a+\iunit x,b+\iunit x,c+\iunit x,d+\iunit x,-n\atop\iunit x-\frac{1}{2}n,1-n-a+\iunit x,1-n-b+\iunit x,1-n-c+\iunit x,1-n-d+\iunit x};1\right)}}}$

Wilson: Special value

${\displaystyle{\displaystyle{\displaystyle\Wilson{n}@{-a^{2}}{a}{b}{c}{d}=\pochhammer{a+b}{n}\pochhammer{a+c}{n}\pochhammer{a+d}{n}}}}$

Uniqueness of orthogonality measure

${\displaystyle{\displaystyle{\displaystyle\frac{|\Wilson{n}@{-a^{2}}{a}{b}{c}{d}|^{2}}{h_{n}}=O(n^{4\realpart{a}-1})\hbox{as $n\to\infty$.}}}}$

<< Orthogonal Polynomials

Wilson

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