DLMF:15.10.E11 (Q5127)

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DLMF:15.10.E11
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    Statements

    test
    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).}}
    0 references
    DLMF id
    DLMF:15.10.E11
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    test
    F ⁡ ( a , b ; c ; z ) Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)}}
    xml-id
    C15.S2.E1.m2afdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1jdec
    0 references
     
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