DLMF:15.10.E12 (Q5128)

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DLMF:15.10.E12
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    w 2 ( z ) = z 1 - c F ( a - c + 1 , b - c + 1 2 - c ; z ) = z 1 - c ( 1 - z ) c - a - b F ( 1 - a , 1 - b 2 - c ; z ) = z 1 - c ( 1 - z ) c - a - 1 F ( a - c + 1 , 1 - b 2 - c ; z z - 1 ) = z 1 - c ( 1 - z ) c - b - 1 F ( 1 - a , b - c + 1 2 - c ; z z - 1 ) . subscript 𝑤 2 𝑧 superscript 𝑧 1 𝑐 Gauss-hypergeometric-F 𝑎 𝑐 1 𝑏 𝑐 1 2 𝑐 𝑧 superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑎 𝑏 Gauss-hypergeometric-F 1 𝑎 1 𝑏 2 𝑐 𝑧 superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑎 1 Gauss-hypergeometric-F 𝑎 𝑐 1 1 𝑏 2 𝑐 𝑧 𝑧 1 superscript 𝑧 1 𝑐 superscript 1 𝑧 𝑐 𝑏 1 Gauss-hypergeometric-F 1 𝑎 𝑏 𝑐 1 2 𝑐 𝑧 𝑧 1 {\displaystyle{\displaystyle w_{2}(z)={z^{1-c}}F\left({a-c+1,b-c+1\atop 2-c};z% \right)={z^{1-c}(1-z)^{c-a-b}}\*F\left({1-a,1-b\atop 2-c};z\right)={z^{1-c}(1-% z)^{c-a-1}}\*F\left({a-c+1,1-b\atop 2-c};\frac{z}{z-1}\right)={z^{1-c}(1-z)^{c% -b-1}}\*F\left({1-a,b-c+1\atop 2-c};\frac{z}{z-1}\right).}}
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