DLMF:15.5.E21 (Q5038)

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DLMF:15.5.E21
No description defined

    Statements

    Defining formula
    c ⁢ ( 1 - z ) ⁢ ( d F ⁡ ( a , b ; c ; z ) / d z ) = ( c - a ) ⁢ ( c - b ) ⁢ F ⁡ ( a , b ; c + 1 ; z ) + c ⁢ ( a + b - c ) ⁢ F ⁡ ( a , b ; c ; z ) . 𝑐 1 𝑧 derivative Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 𝑧 𝑐 𝑎 𝑐 𝑏 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 1 𝑧 𝑐 𝑎 𝑏 𝑐 Gauss-hypergeometric-F 𝑎 𝑏 𝑐 𝑧 {\displaystyle{\displaystyle c(1-z)\left(\ifrac{\mathrm{d}F\left(a,b;c;z\right% )}{\mathrm{d}z}\right)=(c-a)(c-b)F\left(a,b;c+1;z\right)+c(a+b-c)F\left(a,b;c;% z\right).}}
    0 references
    DLMF id
    DLMF:15.5.E21
    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.m2asdec
    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.m2akdec
    0 references
    Q11644
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C15.S1.XMD3.m1tdec
    0 references
    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1sdec
    0 references
    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1sdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1sdec
    0 references
     
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