DLMF:8.17.E9 (Q2649)

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DLMF:8.17.E9
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    Statements

    Defining formula
    B x ⁡ ( a , b ) = x a ⁢ ( 1 - x ) b - 1 a ⁢ F ⁡ ( 1 , 1 - b a + 1 ; x x - 1 ) . incomplete-Beta 𝑥 𝑎 𝑏 superscript 𝑥 𝑎 superscript 1 𝑥 𝑏 1 𝑎 Gauss-hypergeometric-F 1 1 𝑏 𝑎 1 𝑥 𝑥 1 {\displaystyle{\displaystyle\mathrm{B}_{x}\left(a,b\right)=\frac{x^{a}(1-x)^{b% -1}}{a}F\left({1,1-b\atop a+1};\frac{x}{x-1}\right).}}
    0 references
    DLMF id
    DLMF:8.17.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.m2abdec
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    incomplete beta function
    Defining formula
    B x ⁡ ( a , b ) incomplete-Beta 𝑥 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}_{\NVar{x}}\left(\NVar{a},\NVar{b}\right% )}}
    xml-id
    C8.S17.E1.m2addec
    0 references
    Q11360
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C8.S17.XMD1.m1gdec
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    Q11361
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C8.S17.XMD2.m1fdec
    0 references
    Q11362
    Defining formula
    x 𝑥 {\displaystyle{\displaystyle x}}
    xml-id
    C8.S17.XMD3.m1gdec
    0 references
     
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