DLMF:15.8.E6 (Q5063)

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DLMF:15.8.E6
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    Statements

    test
    F ⁡ ( - m , b c ; z ) = ( b ) m ( c ) m ⁢ ( - z ) m ⁢ F ⁡ ( - m , 1 - c - m 1 - b - m ; 1 z ) = ( b ) m ( c ) m ⁢ ( 1 - z ) m ⁢ F ⁡ ( - m , c - b 1 - b - m ; 1 1 - z ) , Gauss-hypergeometric-F 𝑚 𝑏 𝑐 𝑧 subscript 𝑏 𝑚 subscript 𝑐 𝑚 superscript 𝑧 𝑚 Gauss-hypergeometric-F 𝑚 1 𝑐 𝑚 1 𝑏 𝑚 1 𝑧 subscript 𝑏 𝑚 subscript 𝑐 𝑚 superscript 1 𝑧 𝑚 Gauss-hypergeometric-F 𝑚 𝑐 𝑏 1 𝑏 𝑚 1 1 𝑧 {\displaystyle{\displaystyle F\left({-m,b\atop c};z\right)=\frac{(b)_{m}}{(c)_% {m}}(-z)^{m}F\left({-m,1-c-m\atop 1-b-m};\frac{1}{z}\right)=\frac{(b)_{m}}{(c)% _{m}}(1-z)^{m}F\left({-m,c-b\atop 1-b-m};\frac{1}{1-z}\right),}}
    0 references
    DLMF id
    DLMF:15.8.E6
    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.m2adec
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    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1edec
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    Q11646
    test
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1edec
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    Q11647
    test
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1edec
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    Q11651
    test
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C15.S1.XMD9.m1dec
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