DLMF:15.5.E9 (Q5026)

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DLMF:15.5.E9
No description defined

    Statements

    Defining formula
    d n d z n ⁡ ( z c - 1 ⁢ ( 1 - z ) a + b - c ⁢ F ⁡ ( a , b ; c ; z ) ) = ( c - n ) n ⁢ z c - n - 1 ⁢ ( 1 - z ) a + b - c - n ⁢ F ⁡ ( a - n , b - n ; c - n ; z ) . derivative 𝑧 𝑛 superscript 𝑧 𝑐 1 superscript 1 𝑧 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 Pochhammer 𝑐 𝑛 𝑛 superscript 𝑧 𝑐 𝑛 1 superscript 1 𝑧 𝑎 𝑏 𝑐 𝑛 Gauss-hypergeometric-F 𝑎 𝑛 𝑏 𝑛 𝑐 𝑛 𝑧 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(z^% {c-1}(1-z)^{a+b-c}F\left(a,b;c;z\right)\right)={\left(c-n\right)_{n}}z^{c-n-1}% (1-z)^{a+b-c-n}\*F\left(a-n,b-n;c-n;z\right).}}
    0 references
    DLMF id
    DLMF:15.5.E9
    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.m2ahdec
    0 references
    Q10759
    Defining formula
    ( a ) n Pochhammer 𝑎 𝑛 {\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
    xml-id
    C5.S2.SS3.m1agdec
    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.m2ahdec
    0 references
    Q11650
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1gdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1hdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1hdec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1hdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1hdec
    0 references
     
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