Formula:KLS:01.05:08

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∑ n = - ∞ ∞ Γ ⁡ ( n + a ) ⁢ Γ ⁡ ( n + b ) Γ ⁡ ( n + c ) ⁢ Γ ⁡ ( n + d ) = Γ ⁡ ( a ) ⁢ Γ ⁡ ( 1 - a ) ⁢ Γ ⁡ ( b ) ⁢ Γ ⁡ ( 1 - b ) ⁢ Γ ⁡ ( c + d - a - b - 1 ) Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( d - a ) ⁢ Γ ⁡ ( c - b ) ⁢ Γ ⁡ ( d - b ) superscript subscript 𝑛 Euler-Gamma 𝑛 𝑎 Euler-Gamma 𝑛 𝑏 Euler-Gamma 𝑛 𝑐 Euler-Gamma 𝑛 𝑑 Euler-Gamma 𝑎 Euler-Gamma 1 𝑎 Euler-Gamma 𝑏 Euler-Gamma 1 𝑏 Euler-Gamma 𝑐 𝑑 𝑎 𝑏 1 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑑 𝑎 Euler-Gamma 𝑐 𝑏 Euler-Gamma 𝑑 𝑏 {\displaystyle{\displaystyle{\displaystyle{}{}\sum_{n=-\infty}^{\infty}\frac{% \Gamma\left(n+a\right)\Gamma\left(n+b\right)}{\Gamma\left(n+c\right)\Gamma% \left(n+d\right)}=\frac{\Gamma\left(a\right)\Gamma\left(1-a\right)\Gamma\left(% b\right)\Gamma\left(1-b\right)\Gamma\left(c+d-a-b-1\right)}{\Gamma\left(c-a% \right)\Gamma\left(d-a\right)\Gamma\left(c-b\right)\Gamma\left(d-b\right)}}}}

Proof

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Symbols List

Σ Σ {\displaystyle{\displaystyle{\displaystyle\Sigma}}}  : sum : http://drmf.wmflabs.org/wiki/Definition:sum
Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1

Bibliography

Equation in Section 1.5 of KLS.

URL links

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