Wilson

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Wilson

Hypergeometric representation

W n ⁡ ( x 2 ; a , b , c , d ) ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n = \HyperpFq ⁢ 43 ⁢ @ ⁢ @ - n , n + a + b + c + d - 1 , a + i ⁢ x , a - i ⁢ x ⁢ a + b , a + c , a + d ⁢ 1 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 \HyperpFq 43 @ @ 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑎 imaginary-unit 𝑥 𝑎 imaginary-unit 𝑥 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 1 {\displaystyle{\displaystyle{\displaystyle\frac{W_{n}\!\left(x^{2};a,b,c,d% \right)}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{\left(a+d\right)_{n}}}{}% =\HyperpFq{4}{3}@@{-n,n+a+b+c+d-1,a+\mathrm{i}x,a-\mathrm{i}x}{a+b,a+c,a+d}{1}% }}} {\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)

1 2 ⁢ π ⁢ ∫ 0 ∞ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) ⁢ Γ ⁡ ( d + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 ⁢ W m ⁡ ( x 2 ; a , b , c , d ) ⁢ W n ⁡ ( x 2 ; a , b , c , d ) ⁢ 𝑑 x = Γ ⁡ ( n + a + b ) ⁢ ⋯ ⁢ Γ ⁡ ( n + c + d ) Γ ⁡ ( 2 ⁢ n + a + b + c + d ) ⁢ ( n + a + b + c + d - 1 ) n ⁢ n ! ⁢ δ m , n 1 2 superscript subscript 0 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 Wilson-polynomial-W 𝑚 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 differential-d 𝑥 Euler-Gamma 𝑛 𝑎 𝑏 ⋯ Euler-Gamma 𝑛 𝑐 𝑑 Euler-Gamma 2 𝑛 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑛 Kronecker-delta 𝑚 𝑛 {\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(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}}}} {\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} }
Γ ⁡ ( n + a + b ) ⁢ ⋯ ⁢ Γ ⁡ ( n + c + d ) = Γ ⁡ ( n + a + b ) ⁢ Γ ⁡ ( n + a + c ) ⁢ Γ ⁡ ( n + a + d ) ⁢ Γ ⁡ ( n + b + c ) ⁢ Γ ⁡ ( n + b + d ) ⁢ Γ ⁡ ( n + c + d ) Euler-Gamma 𝑛 𝑎 𝑏 ⋯ Euler-Gamma 𝑛 𝑐 𝑑 Euler-Gamma 𝑛 𝑎 𝑏 Euler-Gamma 𝑛 𝑎 𝑐 Euler-Gamma 𝑛 𝑎 𝑑 Euler-Gamma 𝑛 𝑏 𝑐 Euler-Gamma 𝑛 𝑏 𝑑 Euler-Gamma 𝑛 𝑐 𝑑 {\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 \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} }
1 2 ⁢ π ⁢ ∫ 0 ∞ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) ⁢ Γ ⁡ ( d + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 ⁢ W m ⁡ ( x 2 ; a , b , c , d ) ⁢ W n ⁡ ( x 2 ; a , b , c , d ) ⁢ 𝑑 x + Γ ⁡ ( a + b ) ⁢ Γ ⁡ ( a + c ) ⁢ Γ ⁡ ( a + d ) ⁢ Γ ⁡ ( b - a ) ⁢ Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( d - a ) Γ ⁡ ( - 2 ⁢ a ) ⁢ ∑ k = 0 , 1 , 2 ⁢ … a + k < 0 ( 2 ⁢ a ) k ⁢ ( a + 1 ) k ⁢ ( a + b ) k ⁢ ( a + c ) k ⁢ ( a + d ) k ( a ) k ⁢ ( a - b + 1 ) k ⁢ ( a - c + 1 ) k ⁢ ( a - d + 1 ) k ⁢ k ! ⁢ W m ⁡ ( - ( a + k ) 2 ; a , b , c , d ) ⁢ W n ⁡ ( - ( a + k ) 2 ; a , b , c , d ) = Γ ⁡ ( n + a + b ) ⁢ ⋯ ⁢ Γ ⁡ ( n + c + d ) Γ ⁡ ( 2 ⁢ n + a + b + c + d ) ⁢ ( n + a + b + c + d - 1 ) n ⁢ n ! ⁢ δ m , n 1 2 superscript subscript 0 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 Wilson-polynomial-W 𝑚 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 differential-d 𝑥 Euler-Gamma 𝑎 𝑏 Euler-Gamma 𝑎 𝑐 Euler-Gamma 𝑎 𝑑 Euler-Gamma 𝑏 𝑎 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑑 𝑎 Euler-Gamma 2 𝑎 subscript 𝑘 0 1 2 … 𝑎 𝑘 0 Pochhammer-symbol 2 𝑎 𝑘 Pochhammer-symbol 𝑎 1 𝑘 Pochhammer-symbol 𝑎 𝑏 𝑘 Pochhammer-symbol 𝑎 𝑐 𝑘 Pochhammer-symbol 𝑎 𝑑 𝑘 Pochhammer-symbol 𝑎 𝑘 Pochhammer-symbol 𝑎 𝑏 1 𝑘 Pochhammer-symbol 𝑎 𝑐 1 𝑘 Pochhammer-symbol 𝑎 𝑑 1 𝑘 𝑘 Wilson-polynomial-W 𝑚 superscript 𝑎 𝑘 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑎 𝑘 2 𝑎 𝑏 𝑐 𝑑 Euler-Gamma 𝑛 𝑎 𝑏 ⋯ Euler-Gamma 𝑛 𝑐 𝑑 Euler-Gamma 2 𝑛 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑛 Kronecker-delta 𝑚 𝑛 {\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}% }}} {\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

- ( a 2 + x 2 ) ⁢ W ~ n ⁡ ( x 2 ) = A n ⁢ W ~ n + 1 ⁡ ( x 2 ) - ( A n + C n ) ⁢ W ~ n ⁡ ( x 2 ) + C n ⁢ W ~ n - 1 ⁡ ( x 2 ) superscript 𝑎 2 superscript 𝑥 2 Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 Wilson-polynomial-normalized-W-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 subscript 𝐶 𝑛 Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 subscript 𝐶 𝑛 Wilson-polynomial-normalized-W-tilde 𝑛 1 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle-\left(a^{2}+x^{2}\right){\tilde{W}}% _{n}\!\left(x^{2}\right)=A_{n}{\tilde{W}}_{n+1}\!\left(x^{2}\right)-\left(A_{n% }+C_{n}\right){\tilde{W}}_{n}\!\left(x^{2}\right)+C_{n}{\tilde{W}}_{n-1}\!% \left(x^{2}\right)}}} {\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): C n = n ⁢ ( n + b + c - 1 ) ⁢ ( n + b + d - 1 ) ⁢ ( n + c + d - 1 ) ( 2 ⁢ n + a + b + c + d - 2 ) ⁢ ( 2 ⁢ n + a + b + c + d - 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝑏 𝑐 1 𝑛 𝑏 𝑑 1 𝑛 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 2 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 {\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)}}}} &
A n = ( n + a + b + c + d - 1 ) ⁢ ( n + a + b ) ⁢ ( n + a + c ) ⁢ ( n + a + d ) ( 2 ⁢ n + a + b + c + d - 1 ) ⁢ ( 2 ⁢ n + a + b + c + d ) subscript 𝐴 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑎 𝑏 𝑛 𝑎 𝑐 𝑛 𝑎 𝑑 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 {\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)}}}}


W ~ n ⁡ ( x 2 ) := W ~ n ⁡ ( x 2 ; a , b , c , d ) = W n ⁡ ( x 2 ; a , b , c , d ) ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n assign Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{W}}_{n}\!\left(x^{2}\right):% ={\tilde{W}}_{n}\!\left(x^{2};a,b,c,d\right)=\frac{W_{n}\!\left(x^{2};a,b,c,d% \right)}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{\left(a+d\right)_{n}}}}}} {\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

x ⁢ W ^ n ⁡ ( x ) = W ^ n + 1 ⁡ ( x ) + ( A n + C n - a 2 ) ⁢ W ^ n ⁡ ( x ) + A n - 1 ⁢ C n ⁢ W ^ n - 1 ⁡ ( x ) 𝑥 Wilson-polynomial-monic 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-monic 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 subscript 𝐶 𝑛 superscript 𝑎 2 Wilson-polynomial-monic 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 subscript 𝐴 𝑛 1 subscript 𝐶 𝑛 Wilson-polynomial-monic 𝑛 1 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle x{\widehat{W}}_{n}\!\left(x\right)=% {\widehat{W}}_{n+1}\!\left(x\right)+(A_{n}+C_{n}-a^{2}){\widehat{W}}_{n}\!% \left(x\right)+A_{n-1}C_{n}{\widehat{W}}_{n-1}\!\left(x\right)}}} {\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): C n = n ⁢ ( n + b + c - 1 ) ⁢ ( n + b + d - 1 ) ⁢ ( n + c + d - 1 ) ( 2 ⁢ n + a + b + c + d - 2 ) ⁢ ( 2 ⁢ n + a + b + c + d - 1 ) subscript 𝐶 𝑛 𝑛 𝑛 𝑏 𝑐 1 𝑛 𝑏 𝑑 1 𝑛 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 2 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 {\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)}}}} &
A n = ( n + a + b + c + d - 1 ) ⁢ ( n + a + b ) ⁢ ( n + a + c ) ⁢ ( n + a + d ) ( 2 ⁢ n + a + b + c + d - 1 ) ⁢ ( 2 ⁢ n + a + b + c + d ) subscript 𝐴 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 𝑎 𝑏 𝑛 𝑎 𝑐 𝑛 𝑎 𝑑 2 𝑛 𝑎 𝑏 𝑐 𝑑 1 2 𝑛 𝑎 𝑏 𝑐 𝑑 {\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)}}}}


W n ⁡ ( x 2 ; a , b , c , d ) = ( - 1 ) n ⁢ ( n + a + b + c + d - 1 ) n ⁢ W ^ n ⁡ ( x 2 ) Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 superscript 1 𝑛 Pochhammer-symbol 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑛 Wilson-polynomial-monic 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle W_{n}\!\left(x^{2};a,b,c,d\right)=(% -1)^{n}{\left(n+a+b+c+d-1\right)_{n}}{\widehat{W}}_{n}\!\left(x^{2}\right)}}} {\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

n ⁢ ( n + a + b + c + d - 1 ) ⁢ y ⁢ ( x ) = B ⁢ ( x ) ⁢ y ⁢ ( x + i ) - [ B ⁢ ( x ) + D ⁢ ( x ) ] ⁢ y ⁢ ( x ) + D ⁢ ( x ) ⁢ y ⁢ ( x - i ) 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 𝑦 𝑥 𝐵 𝑥 𝑦 𝑥 imaginary-unit delimited-[] 𝐵 𝑥 𝐷 𝑥 𝑦 𝑥 𝐷 𝑥 𝑦 𝑥 imaginary-unit {\displaystyle{\displaystyle{\displaystyle n(n+a+b+c+d-1)y(x){}=B(x)y(x+% \mathrm{i})-\left[B(x)+D(x)\right]y(x)+D(x)y(x-\mathrm{i})}}} {\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): D ⁢ ( x ) = ( a + i ⁢ x ) ⁢ ( b + i ⁢ x ) ⁢ ( c + i ⁢ x ) ⁢ ( d + i ⁢ x ) 2 ⁢ i ⁢ x ⁢ ( 2 ⁢ i ⁢ x + 1 ) 𝐷 𝑥 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑑 imaginary-unit 𝑥 2 imaginary-unit 𝑥 2 imaginary-unit 𝑥 1 {\displaystyle{\displaystyle{\displaystyle D(x)=\frac{(a+\mathrm{i}x)(b+% \mathrm{i}x)(c+\mathrm{i}x)(d+\mathrm{i}x)}{2\mathrm{i}x(2\mathrm{i}x+1)}}}} &

B ⁢ ( x ) = ( a - i ⁢ x ) ⁢ ( b - i ⁢ x ) ⁢ ( c - i ⁢ x ) ⁢ ( d - i ⁢ x ) 2 ⁢ i ⁢ x ⁢ ( 2 ⁢ i ⁢ x - 1 ) 𝐵 𝑥 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑑 imaginary-unit 𝑥 2 imaginary-unit 𝑥 2 imaginary-unit 𝑥 1 {\displaystyle{\displaystyle{\displaystyle B(x)=\frac{(a-\mathrm{i}x)(b-% \mathrm{i}x)(c-\mathrm{i}x)(d-\mathrm{i}x)}{2\mathrm{i}x(2\mathrm{i}x-1)}}}} &

y ⁢ ( x ) = W n ⁡ ( x 2 ; a , b , c , d ) 𝑦 𝑥 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle y(x)=W_{n}\!\left(x^{2};a,b,c,d% \right)}}}


Forward shift operator

W n ⁡ ( ( x + 1 2 ⁢ i ) 2 ; a , b , c , d ) - W n ⁡ ( ( x - 1 2 ⁢ i ) 2 ; a , b , c , d ) = - 2 ⁢ i ⁢ n ⁢ x ⁢ ( n + a + b + c + d - 1 ) ⁢ W n - 1 ⁡ ( x 2 ; a + 1 2 , b + 1 2 , c + 1 2 , d + 1 2 ) Wilson-polynomial-W 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑑 2 imaginary-unit 𝑛 𝑥 𝑛 𝑎 𝑏 𝑐 𝑑 1 Wilson-polynomial-W 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle W_{n}\!\left((x+\textstyle\frac{1}{% 2}\mathrm{i})^{2};a,b,c,d\right)-W_{n}\!\left((x-\textstyle\frac{1}{2}\mathrm{% i})^{2};a,b,c,d\right){}=-2\mathrm{i}nx(n+a+b+c+d-1)W_{n-1}\!\left(x^{2};a+% \textstyle\frac{1}{2},b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+% \textstyle\frac{1}{2}\right)}}} {\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}} }
δ ⁢ W n ⁡ ( x 2 ; a , b , c , d ) δ ⁢ x 2 = - n ⁢ ( n + a + b + c + d - 1 ) ⁢ W n - 1 ⁡ ( x 2 ; a + 1 2 , b + 1 2 , c + 1 2 , d + 1 2 ) 𝛿 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 𝛿 superscript 𝑥 2 𝑛 𝑛 𝑎 𝑏 𝑐 𝑑 1 Wilson-polynomial-W 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta W_{n}\!\left(x^{2};a,b,% c,d\right)}{\delta x^{2}}{}=-n(n+a+b+c+d-1)W_{n-1}\!\left(x^{2};a+\textstyle% \frac{1}{2},b+\textstyle\frac{1}{2},c+\textstyle\frac{1}{2},d+\textstyle\frac{% 1}{2}\right)}}} {\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

( a - 1 2 - i ⁢ x ) ⁢ ( b - 1 2 - i ⁢ x ) ⁢ ( c - 1 2 - i ⁢ x ) ⁢ ( d - 1 2 - i ⁢ x ) ⁢ W n ⁡ ( ( x + 1 2 ⁢ i ) 2 ; a , b , c , d ) - ( a - 1 2 + i ⁢ x ) ⁢ ( b - 1 2 + i ⁢ x ) ⁢ ( c - 1 2 + i ⁢ x ) ⁢ ( d - 1 2 + i ⁢ x ) ⁢ W n ⁡ ( ( x - 1 2 ⁢ i ) 2 ; a , b , c , d ) = - 2 ⁢ i ⁢ x ⁢ W n + 1 ⁡ ( x 2 ; a - 1 2 , b - 1 2 , c - 1 2 , d - 1 2 ) 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 𝑑 1 2 imaginary-unit 𝑥 Wilson-polynomial-W 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑑 𝑎 1 2 imaginary-unit 𝑥 𝑏 1 2 imaginary-unit 𝑥 𝑐 1 2 imaginary-unit 𝑥 𝑑 1 2 imaginary-unit 𝑥 Wilson-polynomial-W 𝑛 superscript 𝑥 1 2 imaginary-unit 2 𝑎 𝑏 𝑐 𝑑 2 imaginary-unit 𝑥 Wilson-polynomial-W 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle(a-\textstyle\frac{1}{2}-\mathrm{i}x% )(b-\textstyle\frac{1}{2}-\mathrm{i}x)(c-\textstyle\frac{1}{2}-\mathrm{i}x)(d-% \textstyle\frac{1}{2}-\mathrm{i}x)W_{n}\!\left((x+\textstyle\frac{1}{2}\mathrm% {i})^{2};a,b,c,d\right){}-(a-\textstyle\frac{1}{2}+\mathrm{i}x)(b-\textstyle% \frac{1}{2}+\mathrm{i}x)(c-\textstyle\frac{1}{2}+\mathrm{i}x)(d-\textstyle% \frac{1}{2}+\mathrm{i}x)W_{n}\!\left((x-\textstyle\frac{1}{2}\mathrm{i})^{2};a% ,b,c,d\right){}=-2\mathrm{i}xW_{n+1}\!\left(x^{2};a-\textstyle\frac{1}{2},b-% \textstyle\frac{1}{2},c-\textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}} {\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}} }
δ ⁢ [ ω ⁢ ( x ; a , b , c , d ) ⁢ W n ⁡ ( x 2 ; a , b , c , d ) ] δ ⁢ x 2 = ω ⁢ ( x ; a - 1 2 , b - 1 2 ⁢ c - 1 2 , d - 1 2 ) ⁢ W n + 1 ⁡ ( x 2 ; a - 1 2 , b - 1 2 , c - 1 2 , d - 1 2 ) 𝛿 delimited-[] 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 𝛿 superscript 𝑥 2 𝜔 𝑥 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 Wilson-polynomial-W 𝑛 1 superscript 𝑥 2 𝑎 1 2 𝑏 1 2 𝑐 1 2 𝑑 1 2 {\displaystyle{\displaystyle{\displaystyle\frac{\delta\left[\omega(x;a,b,c,d)W% _{n}\!\left(x^{2};a,b,c,d\right)\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})W_{n+1}\!\left(x^{2};a-\textstyle\frac{1}{2},b-\textstyle\frac{1}{2},c-% \textstyle\frac{1}{2},d-\textstyle\frac{1}{2}\right)}}} {\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): ω ⁢ ( x ; a , b , c , d ) := 1 2 ⁢ i ⁢ x ⁢ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) ⁢ Γ ⁡ ( d + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 assign 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 1 2 imaginary-unit 𝑥 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d):=\frac{1}{2\mathrm% {i}x}\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}}}}


Rodrigues-type formula

ω ⁢ ( x ; a , b , c , d ) ⁢ W n ⁡ ( x 2 ; a , b , c , d ) = ( δ δ ⁢ x 2 ) n ⁢ [ ω ⁢ ( x ; a + 1 2 ⁢ n , b + 1 2 ⁢ n ⁢ c + 1 2 ⁢ n , d + 1 2 ⁢ n ) ] 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 superscript 𝛿 𝛿 superscript 𝑥 2 𝑛 delimited-[] 𝜔 𝑥 𝑎 1 2 𝑛 𝑏 1 2 𝑛 𝑐 1 2 𝑛 𝑑 1 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d)W_{n}\!\left(x^{2};% a,b,c,d\right){}=\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]}}} {\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): ω ⁢ ( x ; a , b , c , d ) := 1 2 ⁢ i ⁢ x ⁢ | Γ ⁡ ( a + i ⁢ x ) ⁢ Γ ⁡ ( b + i ⁢ x ) ⁢ Γ ⁡ ( c + i ⁢ x ) ⁢ Γ ⁡ ( d + i ⁢ x ) Γ ⁡ ( 2 ⁢ i ⁢ x ) | 2 assign 𝜔 𝑥 𝑎 𝑏 𝑐 𝑑 1 2 imaginary-unit 𝑥 superscript Euler-Gamma 𝑎 imaginary-unit 𝑥 Euler-Gamma 𝑏 imaginary-unit 𝑥 Euler-Gamma 𝑐 imaginary-unit 𝑥 Euler-Gamma 𝑑 imaginary-unit 𝑥 Euler-Gamma 2 imaginary-unit 𝑥 2 {\displaystyle{\displaystyle{\displaystyle\omega(x;a,b,c,d):=\frac{1}{2\mathrm% {i}x}\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}}}}


Generating functions

\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ a + i ⁢ x , b + i ⁢ x ⁢ a + b ⁢ t ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ c - i ⁢ x , d - i ⁢ x ⁢ c + d ⁢ t = ∑ n = 0 ∞ W n ⁡ ( x 2 ; a , b , c , d ) ⁢ t n ( a + b ) n ⁢ ( c + d ) n ⁢ n ! \HyperpFq 21 @ @ 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 𝑎 𝑏 𝑡 \HyperpFq 21 @ @ 𝑐 imaginary-unit 𝑥 𝑑 imaginary-unit 𝑥 𝑐 𝑑 𝑡 superscript subscript 𝑛 0 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 superscript 𝑡 𝑛 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑐 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{a+\mathrm{i}x,b+% \mathrm{i}x}{a+b}{t}\,\HyperpFq{2}{1}@@{c-\mathrm{i}x,d-\mathrm{i}x}{c+d}{t}=% \sum_{n=0}^{\infty}\frac{W_{n}\!\left(x^{2};a,b,c,d\right)t^{n}}{{\left(a+b% \right)_{n}}{\left(c+d\right)_{n}}n!}}}} {\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!} }
\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ a + i ⁢ x , c + i ⁢ x ⁢ a + c ⁢ t ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ b - i ⁢ x , d - i ⁢ x ⁢ b + d ⁢ t = ∑ n = 0 ∞ W n ⁡ ( x 2 ; a , b , c , d ) ⁢ t n ( a + c ) n ⁢ ( b + d ) n ⁢ n ! \HyperpFq 21 @ @ 𝑎 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑎 𝑐 𝑡 \HyperpFq 21 @ @ 𝑏 imaginary-unit 𝑥 𝑑 imaginary-unit 𝑥 𝑏 𝑑 𝑡 superscript subscript 𝑛 0 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 superscript 𝑡 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑏 𝑑 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{a+\mathrm{i}x,c+% \mathrm{i}x}{a+c}{t}\,\HyperpFq{2}{1}@@{b-\mathrm{i}x,d-\mathrm{i}x}{b+d}{t}=% \sum_{n=0}^{\infty}\frac{W_{n}\!\left(x^{2};a,b,c,d\right)t^{n}}{{\left(a+c% \right)_{n}}{\left(b+d\right)_{n}}n!}}}} {\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!} }
\HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ a + i ⁢ x , d + i ⁢ x ⁢ a + d ⁢ t ⁢ \HyperpFq ⁢ 21 ⁢ @ ⁢ @ ⁢ b - i ⁢ x , c - i ⁢ x ⁢ b + c ⁢ t = ∑ n = 0 ∞ W n ⁡ ( x 2 ; a , b , c , d ) ⁢ t n ( a + d ) n ⁢ ( b + c ) n ⁢ n ! \HyperpFq 21 @ @ 𝑎 imaginary-unit 𝑥 𝑑 imaginary-unit 𝑥 𝑎 𝑑 𝑡 \HyperpFq 21 @ @ 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑏 𝑐 𝑡 superscript subscript 𝑛 0 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 superscript 𝑡 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 Pochhammer-symbol 𝑏 𝑐 𝑛 𝑛 {\displaystyle{\displaystyle{\displaystyle\HyperpFq{2}{1}@@{a+\mathrm{i}x,d+% \mathrm{i}x}{a+d}{t}\,\HyperpFq{2}{1}@@{b-\mathrm{i}x,c-\mathrm{i}x}{b+c}{t}=% \sum_{n=0}^{\infty}\frac{W_{n}\!\left(x^{2};a,b,c,d\right)t^{n}}{{\left(a+d% \right)_{n}}{\left(b+c\right)_{n}}n!}}}} {\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!} }
( 1 - t ) 1 - a - b - c - d ⁢ \HyperpFq ⁢ 43 ⁢ @ ⁢ @ ⁢ 1 2 ⁢ ( a + b + c + d - 1 ) , 1 2 ⁢ ( a + b + c + d ) , a + i ⁢ x , a - i ⁢ x ⁢ a + b , a + c , a + d - 4 ⁢ t ( 1 - t ) 2 = ∑ n = 0 ∞ ( a + b + c + d - 1 ) n ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n ⁢ n ! ⁢ W n ⁡ ( x 2 ; a , b , c , d ) ⁢ t n superscript 1 𝑡 1 𝑎 𝑏 𝑐 𝑑 \HyperpFq 43 @ @ 1 2 𝑎 𝑏 𝑐 𝑑 1 1 2 𝑎 𝑏 𝑐 𝑑 𝑎 imaginary-unit 𝑥 𝑎 imaginary-unit 𝑥 𝑎 𝑏 𝑎 𝑐 𝑎 𝑑 4 𝑡 superscript 1 𝑡 2 superscript subscript 𝑛 0 Pochhammer-symbol 𝑎 𝑏 𝑐 𝑑 1 𝑛 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 𝑛 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 superscript 𝑡 𝑛 {\displaystyle{\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+\mathrm{i}x,a-\mathrm{i}x}{a+b,% a+c,a+d}{-\frac{4t}{(1-t)^{2}}}{}=\sum_{n=0}^{\infty}\frac{{\left(a+b+c+d-1% \right)_{n}}}{{\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{\left(a+d\right)_{n% }}n!}W_{n}\!\left(x^{2};a,b,c,d\right)t^{n}}}} {\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

lim d → ∞ ⁡ W n ⁡ ( x 2 ; a , b , c , d ) ( a + d ) n = S n ⁡ ( x 2 ; a , b , c ) subscript → 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑑 𝑛 continuous-dual-Hahn-normalized-S 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 {\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)}}} {\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

lim t → ∞ ⁡ W n ⁡ ( ( x + t ) 2 ; a - i ⁢ t , b - i ⁢ t , c + i ⁢ t , d + i ⁢ t ) ( - 2 ⁢ t ) n ⁢ n ! = p n ⁡ ( x ; a , b , c , d ) subscript → 𝑡 Wilson-polynomial-W 𝑛 superscript 𝑥 𝑡 2 𝑎 imaginary-unit 𝑡 𝑏 imaginary-unit 𝑡 𝑐 imaginary-unit 𝑡 𝑑 imaginary-unit 𝑡 superscript 2 𝑡 𝑛 𝑛 continuous-Hahn-polynomial 𝑛 𝑥 𝑎 𝑏 𝑐 𝑑 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{W_{n}% \!\left((x+t)^{2};a-\mathrm{i}t,b-\mathrm{i}t,c+\mathrm{i}t,d+\mathrm{i}t% \right)}{(-2t)^{n}n!}=p_{n}\!\left(x;a,b,c,d\right)}}} {\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

lim t → ∞ ⁡ W n ⁡ ( 1 2 ⁢ ( 1 - x ) ⁢ t 2 ; 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ ( α + 1 ) , 1 2 ⁢ ( β + 1 ) + i ⁢ t , 1 2 ( β + 1 ) - i t ) t 2 ⁢ n ⁢ n ! = P n ( α , β ) ⁡ ( x ) subscript → 𝑡 fragments Wilson-polynomial-W 𝑛 1 2 1 𝑥 superscript 𝑡 2 1 2 𝛼 1 1 2 𝛼 1 1 2 𝛽 1 imaginary-unit 𝑡 fragments 1 2 fragments ( β 1 imaginary-unit t ) superscript 𝑡 2 𝑛 𝑛 Jacobi-polynomial-P 𝛼 𝛽 𝑛 𝑥 {\displaystyle{\displaystyle{\displaystyle\lim_{t\rightarrow\infty}\frac{W_{n}% \!\left(\frac{1}{2}(1-x)t^{2};\frac{1}{2}(\alpha+1),\frac{1}{2}(\alpha+1),% \frac{1}{2}(\beta+1)+\mathrm{i}t,\frac{1}{2}(\beta+1\right)-\mathrm{i}t)}{t^{2% n}n!}{}=P^{(\alpha,\beta)}_{n}\left(x\right)}}} {\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

( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n ( a + b ) k ⁢ ( a + c ) k ⁢ ( a + d ) k = ( a + b + k ) n - k ⁢ ( a + c + k ) n - k ⁢ ( a + d + k ) n - k Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 Pochhammer-symbol 𝑎 𝑏 𝑘 Pochhammer-symbol 𝑎 𝑐 𝑘 Pochhammer-symbol 𝑎 𝑑 𝑘 Pochhammer-symbol 𝑎 𝑏 𝑘 𝑛 𝑘 Pochhammer-symbol 𝑎 𝑐 𝑘 𝑛 𝑘 Pochhammer-symbol 𝑎 𝑑 𝑘 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\frac{{\left(a+b\right)_{n}}{\left(a% +c\right)_{n}}{\left(a+d\right)_{n}}}{{\left(a+b\right)_{k}}{\left(a+c\right)_% {k}}{\left(a+d\right)_{k}}}={\left(a+b+k\right)_{n-k}}{\left(a+c+k\right)_{n-k% }}{\left(a+d+k\right)_{n-k}}}}} {\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} }
a = 1 2 ⁢ ( γ + δ + 1 ) 𝑎 1 2 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle a=\textstyle\frac{1}{2}(\gamma+% \delta+1)}}} {\displaystyle a=\textstyle\frac{1}{2}(\gamma+\delta+1) }
b = 1 2 ⁢ ( 2 ⁢ α - γ - δ + 1 ) 𝑏 1 2 2 𝛼 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle b=\textstyle\frac{1}{2}(2\alpha-% \gamma-\delta+1)}}} {\displaystyle b=\textstyle\frac{1}{2}(2\alpha-\gamma-\delta+1) }
c = 1 2 ⁢ ( 2 ⁢ β - γ + δ + 1 ) 𝑐 1 2 2 𝛽 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle c=\textstyle\frac{1}{2}(2\beta-% \gamma+\delta+1)}}} {\displaystyle c=\textstyle\frac{1}{2}(2\beta-\gamma+\delta+1) }
d = 1 2 ⁢ ( γ - δ + 1 ) 𝑑 1 2 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle d=\textstyle\frac{1}{2}(\gamma-% \delta+1)}}} {\displaystyle d=\textstyle\frac{1}{2}(\gamma-\delta+1) }
i ⁢ x → x + 1 2 ⁢ ( γ + δ + 1 ) → imaginary-unit 𝑥 𝑥 1 2 𝛾 𝛿 1 {\displaystyle{\displaystyle{\displaystyle\mathrm{i}x\rightarrow x+\textstyle% \frac{1}{2}(\gamma+\delta+1)}}} {\displaystyle \iunit x\rightarrow x+\textstyle\frac{1}{2}(\gamma+\delta+1) }
W ~ n ⁡ ( x 2 ; a , b , c , d ) = W n ⁡ ( x 2 ; a , b , c , d ) ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n Wilson-polynomial-normalized-W-tilde 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle{\tilde{W}}_{n}\!\left(x^{2};a,b,c,d% \right)=\frac{W_{n}\!\left(x^{2};a,b,c,d\right)}{{\left(a+b\right)_{n}}{\left(% a+c\right)_{n}}{\left(a+d\right)_{n}}}}}} {\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}} }
α + 1 = - N   or   β + δ + 1 = - N   or   γ + 1 = - N formulae-sequence 𝛼 1 𝑁 or formulae-sequence 𝛽 𝛿 1 𝑁 or 𝛾 1 𝑁 {\displaystyle{\displaystyle{\displaystyle\alpha+1=-N\quad\textrm{or}\quad% \beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N}}} {\displaystyle \alpha+1=-N\quad\textrm{or}\quad\beta+\delta+1=-N\quad\textrm{or}\quad\gamma+1=-N }

Koornwinder Addendum: Wilson

Hypergeometric representation

W n ⁡ ( x 2 ; a , b , c , d ) = ( a - i ⁢ x ) n ⁢ ( b - i ⁢ x ) n ⁢ ( c - i ⁢ x ) n ⁢ ( d - i ⁢ x ) n ( - 2 ⁢ i ⁢ x ) n ⁢ \HyperpFq ⁢ 76 ⁢ @ ⁢ @ ⁢ 2 ⁢ i ⁢ x - n , i ⁢ x - 1 2 ⁢ n + 1 , a + i ⁢ x , b + i ⁢ x , c + i ⁢ x , d + i ⁢ x , - n ⁢ i ⁢ x - 1 2 ⁢ n , 1 - n - a + i ⁢ x , 1 - n - b + i ⁢ x , 1 - n - c + i ⁢ x , 1 - n - d + i ⁢ x ⁢ 1 Wilson-polynomial-W 𝑛 superscript 𝑥 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 imaginary-unit 𝑥 𝑛 Pochhammer-symbol 𝑏 imaginary-unit 𝑥 𝑛 Pochhammer-symbol 𝑐 imaginary-unit 𝑥 𝑛 Pochhammer-symbol 𝑑 imaginary-unit 𝑥 𝑛 Pochhammer-symbol 2 imaginary-unit 𝑥 𝑛 \HyperpFq 76 @ @ 2 imaginary-unit 𝑥 𝑛 imaginary-unit 𝑥 1 2 𝑛 1 𝑎 imaginary-unit 𝑥 𝑏 imaginary-unit 𝑥 𝑐 imaginary-unit 𝑥 𝑑 imaginary-unit 𝑥 𝑛 imaginary-unit 𝑥 1 2 𝑛 1 𝑛 𝑎 imaginary-unit 𝑥 1 𝑛 𝑏 imaginary-unit 𝑥 1 𝑛 𝑐 imaginary-unit 𝑥 1 𝑛 𝑑 imaginary-unit 𝑥 1 {\displaystyle{\displaystyle{\displaystyle W_{n}\!\left(x^{2};a,b,c,d\right)=% \frac{{\left(a-\mathrm{i}x\right)_{n}}{\left(b-\mathrm{i}x\right)_{n}}{\left(c% -\mathrm{i}x\right)_{n}}{\left(d-\mathrm{i}x\right)_{n}}}{{\left(-2\mathrm{i}x% \right)_{n}}}\HyperpFq{7}{6}@@{2\mathrm{i}x-n,\mathrm{i}x-\frac{1}{2}n+1,a+% \mathrm{i}x,b+\mathrm{i}x,c+\mathrm{i}x,d+\mathrm{i}x,-n}{\mathrm{i}x-\frac{1}% {2}n,1-n-a+\mathrm{i}x,1-n-b+\mathrm{i}x,1-n-c+\mathrm{i}x,1-n-d+\mathrm{i}x}{% 1}}}} {\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} }

Wilson: Special value

W n ⁡ ( - a 2 ; a , b , c , d ) = ( a + b ) n ⁢ ( a + c ) n ⁢ ( a + d ) n Wilson-polynomial-W 𝑛 superscript 𝑎 2 𝑎 𝑏 𝑐 𝑑 Pochhammer-symbol 𝑎 𝑏 𝑛 Pochhammer-symbol 𝑎 𝑐 𝑛 Pochhammer-symbol 𝑎 𝑑 𝑛 {\displaystyle{\displaystyle{\displaystyle W_{n}\!\left(-a^{2};a,b,c,d\right)=% {\left(a+b\right)_{n}}{\left(a+c\right)_{n}}{\left(a+d\right)_{n}}}}} {\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

| W n ⁡ ( - a 2 ; a , b , c , d ) | 2 h n = O ⁢ ( n 4 ⁢ ℜ ⁡ a - 1 ) ⁢ as  n → ∞ . superscript Wilson-polynomial-W 𝑛 superscript 𝑎 2 𝑎 𝑏 𝑐 𝑑 2 subscript ℎ 𝑛 𝑂 superscript 𝑛 4 𝑎 1 as  n → ∞ . {\displaystyle{\displaystyle{\displaystyle\frac{|W_{n}\!\left(-a^{2};a,b,c,d% \right)|^{2}}{h_{n}}=O(n^{4\Re{a}-1})\hbox{as $n\to\infty$.}}}} {\displaystyle \frac{|\Wilson{n}@{-a^2}{a}{b}{c}{d}|^2}{h_n} = O(n^{4\realpart{a}-1}) \hbox{as $n\to\infty$.} }

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