DLMF:15.10.E27 (Q5143)

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DLMF:15.10.E27
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    w 3 ( z ) = e - a π i Γ ( a + b - c + 1 ) Γ ( b - a ) Γ ( b ) Γ ( b - c + 1 ) w 5 ( z ) + e - b π i Γ ( a + b - c + 1 ) Γ ( a - b ) Γ ( a ) Γ ( a - c + 1 ) w 6 ( z ) , subscript 𝑤 3 𝑧 superscript 𝑒 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑎 𝑏 𝑐 1 Euler-Gamma 𝑏 𝑎 Euler-Gamma 𝑏 Euler-Gamma 𝑏 𝑐 1 subscript 𝑤 5 𝑧 superscript 𝑒 𝑏 𝜋 imaginary-unit Euler-Gamma 𝑎 𝑏 𝑐 1 Euler-Gamma 𝑎 𝑏 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑐 1 subscript 𝑤 6 𝑧 {\displaystyle{\displaystyle w_{3}(z)=e^{-a\pi\mathrm{i}}\frac{\Gamma\left(a+b% -c+1\right)\Gamma\left(b-a\right)}{\Gamma\left(b\right)\Gamma\left(b-c+1\right% )}w_{5}(z)+e^{-b\pi\mathrm{i}}\frac{\Gamma\left(a+b-c+1\right)\Gamma\left(a-b% \right)}{\Gamma\left(a\right)\Gamma\left(a-c+1\right)}w_{6}(z),}}
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