Formula:KLS:09.01:01

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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}% }}}

Proof

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Symbols List

W n subscript 𝑊 𝑛 {\displaystyle{\displaystyle{\displaystyle W_{n}}}}  : Wilson polynomial : http://dlmf.nist.gov/18.25#T1.t1.r2
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i

Bibliography

Equation in Section 9.1 of KLS.

URL links

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