DLMF:15.10.E11 (Q5127)

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DLMF:15.10.E11
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    w 1 ( z ) = F ( a , b c ; z ) = ( 1 - z ) c - a - b F ( c - a , c - b c ; z ) = ( 1 - z ) - a F ( a , c - b c ; z z - 1 ) = ( 1 - z ) - b F ( c - a , b c ; z z - 1 ) . subscript 𝑤 1 𝑧 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript 1 𝑧 𝑐 𝑎 𝑏 Gauss-hypergeometric-F 𝑐 𝑎 𝑐 𝑏 𝑐 𝑧 superscript 1 𝑧 𝑎 Gauss-hypergeometric-F 𝑎 𝑐 𝑏 𝑐 𝑧 𝑧 1 superscript 1 𝑧 𝑏 Gauss-hypergeometric-F 𝑐 𝑎 𝑏 𝑐 𝑧 𝑧 1 {\displaystyle{\displaystyle w_{1}(z)=F\left({a,b\atop c};z\right)=(1-z)^{c-a-% b}F\left({c-a,c-b\atop c};z\right)=(1-z)^{-a}F\left({a,c-b\atop c};\frac{z}{z-% 1}\right)=(1-z)^{-b}F\left({c-a,b\atop c};\frac{z}{z-1}\right).}}
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