Formula:KLS:09.01:36

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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}}}}

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
i i {\displaystyle{\displaystyle{\displaystyle\mathrm{i}}}}  : imaginary unit : http://dlmf.nist.gov/1.9.i
F q p subscript subscript 𝐹 𝑞 𝑝 {\displaystyle{\displaystyle{\displaystyle{{}_{p}F_{q}}}}}  : generalized hypergeometric function : http://dlmf.nist.gov/16.2#E1

Bibliography

Equation in Section 9.1 of KLS.

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