Wilson

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Wilson

Hypergeometric representation

Orthogonality relation(s)



Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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

Substitution(s): &


Monic recurrence relation

Substitution(s): &


Difference equation

Substitution(s): &

&


Forward shift operator


Backward shift operator


Substitution(s):


Rodrigues-type formula

Substitution(s):


Generating functions




Limit relations

Wilson polynomial to Continuous dual Hahn polynomial


Wilson polynomial to Continuous Hahn polynomial


Wilson polynomial to Jacobi polynomial

Remarks








Koornwinder Addendum: Wilson

Hypergeometric representation


Wilson: Special value


Uniqueness of orthogonality measure