DLMF:15.5.E7 (Q5024)

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DLMF:15.5.E7
No description defined

    Statements

    Defining formula
    ( ( 1 - z ) ⁢ d d z ⁡ ( 1 - z ) ) n ⁢ ( ( 1 - z ) a - 1 ⁢ F ⁡ ( a , b ; c ; z ) ) = ( - 1 ) n ⁢ ( a ) n ⁢ ( c - b ) n ( c ) n ⁢ ( 1 - z ) a + n - 1 ⁢ F ⁡ ( a + n , b ; c + n ; z ) . superscript 1 𝑧 derivative 𝑧 1 𝑧 𝑛 superscript 1 𝑧 𝑎 1 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 superscript 1 𝑛 Pochhammer 𝑎 𝑛 Pochhammer 𝑐 𝑏 𝑛 Pochhammer 𝑐 𝑛 superscript 1 𝑧 𝑎 𝑛 1 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\left((1-z)\frac{\mathrm{d}}{\mathrm{d}z}(1-z)% \right)^{n}\left((1-z)^{a-1}F\left(a,b;c;z\right)\right)=(-1)^{n}\frac{{\left(% a\right)_{n}}{\left(c-b\right)_{n}}}{{\left(c\right)_{n}}}(1-z)^{a+n-1}\*F% \left(a+n,b;c+n;z\right).}}
    0 references
    DLMF id
    DLMF:15.5.E7
    0 references
    Symbols used
    $$={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$$ Gauss’ hypergeometric function
    Defining formula
    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
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1aedec
    0 references
    derivative of $$f$$ with respect to $$x$$
    Defining formula
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2afdec
    0 references
    Q11650
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1edec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1fdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1fdec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1fdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1fdec
    0 references
     
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