Formula:KLS:09.01:24

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( 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}}}}

Proof

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Symbols List

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
Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
( a ) n subscript 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle(a)_{n}}}}  : Pochhammer symbol : http://dlmf.nist.gov/5.2#iii
W n subscript 𝑊 𝑛 {\displaystyle{\displaystyle{\displaystyle W_{n}}}}  : Wilson polynomial : http://dlmf.nist.gov/18.25#T1.t1.r2

Bibliography

Equation in Section 9.1 of KLS.

URL links

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