DLMF:15.10.E23 (Q5139)

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DLMF:15.10.E23
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    w 5 ( z ) = e a π i Γ ( a - b + 1 ) Γ ( c - a - b ) Γ ( 1 - b ) Γ ( c - b ) w 3 ( z ) + e ( c - b ) π i Γ ( a - b + 1 ) Γ ( a + b - c ) Γ ( a ) Γ ( a - c + 1 ) w 4 ( z ) , subscript 𝑤 5 𝑧 superscript 𝑒 𝑎 𝜋 imaginary-unit Euler-Gamma 𝑎 𝑏 1 Euler-Gamma 𝑐 𝑎 𝑏 Euler-Gamma 1 𝑏 Euler-Gamma 𝑐 𝑏 subscript 𝑤 3 𝑧 superscript 𝑒 𝑐 𝑏 𝜋 imaginary-unit Euler-Gamma 𝑎 𝑏 1 Euler-Gamma 𝑎 𝑏 𝑐 Euler-Gamma 𝑎 Euler-Gamma 𝑎 𝑐 1 subscript 𝑤 4 𝑧 {\displaystyle{\displaystyle w_{5}(z)=e^{a\pi\mathrm{i}}\frac{\Gamma\left(a-b+% 1\right)\Gamma\left(c-a-b\right)}{\Gamma\left(1-b\right)\Gamma\left(c-b\right)% }w_{3}(z)+e^{(c-b)\pi\mathrm{i}}\frac{\Gamma\left(a-b+1\right)\Gamma\left(a+b-% c\right)}{\Gamma\left(a\right)\Gamma\left(a-c+1\right)}w_{4}(z),}}
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