DLMF:15.5.E5 (Q5022)

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DLMF:15.5.E5
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    Statements

    test
    ( z ⁢ d d z ⁡ z ) n ⁢ ( z c - a - 1 ⁢ ( 1 - z ) a + b - c ⁢ F ⁡ ( a , b ; c ; z ) ) = ( c - a ) n ⁢ z c - a + n - 1 ⁢ ( 1 - z ) a - n + b - c ⁢ F ⁡ ( a - n , b ; c ; z ) . superscript 𝑧 derivative 𝑧 𝑧 𝑛 superscript 𝑧 𝑐 𝑎 1 superscript 1 𝑧 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑐 𝑎 𝑛 superscript 𝑧 𝑐 𝑎 𝑛 1 superscript 1 𝑧 𝑎 𝑛 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle\left(z\frac{\mathrm{d}}{\mathrm{d}z}z\right)^{n}% \left(z^{c-a-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-a\right)_{n}% }z^{c-a+n-1}(1-z)^{a-n+b-c}\*F\left(a-n,b;c;z\right).}}
    0 references
    DLMF id
    DLMF:15.5.E5
    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.m2addec
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    Q10759
    test
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1acdec
    0 references
    derivative of $$f$$ with respect to $$x$$
    test
    d f d x derivative 𝑓 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}}}
    xml-id
    C1.S4.E4.m2addec
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    Q11650
    test
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1cdec
    0 references
    Q11644
    test
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1ddec
    0 references
     
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