DLMF:19.16.E24 (Q6342)

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DLMF:19.16.E24
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    Statements

    Defining formula
    R - a ⁡ ( 𝐛 ; 𝐳 ) = z 1 a ′ - b 1 B ⁡ ( b 1 , a ′ - b 1 ) ⁢ ∫ 0 ∞ t b 1 - 1 ⁢ ( t + z 1 ) - a ′ ⁢ R - a ⁡ ( 𝐛 ; 0 , t + z 2 , … , t + z n ) ⁢ d t , Carlson-integral-R 𝑎 𝐛 𝐳 superscript subscript 𝑧 1 superscript 𝑎 ′ subscript 𝑏 1 Euler-Beta subscript 𝑏 1 superscript 𝑎 ′ subscript 𝑏 1 superscript subscript 0 superscript 𝑡 subscript 𝑏 1 1 superscript 𝑡 subscript 𝑧 1 superscript 𝑎 ′ Carlson-integral-R 𝑎 𝐛 0 𝑡 subscript 𝑧 2 … 𝑡 subscript 𝑧 𝑛 𝑡 {\displaystyle{\displaystyle R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{z_% {1}^{a^{\prime}-b_{1}}}{\mathrm{B}\left(b_{1},a^{\prime}-b_{1}\right)}\int_{0}% ^{\infty}t^{b_{1}-1}(t+z_{1})^{-a^{\prime}}\*R_{-a}\left(\mathbf{b};0,t+z_{2},% \dots,t+z_{n}\right)\mathrm{d}t,}}
    0 references
    DLMF id
    DLMF:19.16.E24
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    constraint
    a + a ′ > b 1 > 0 𝑎 superscript 𝑎 ′ subscript 𝑏 1 0 {\displaystyle{\displaystyle a+a^{\prime}>b_{1}>0}}
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    Symbols used
    multivariate hypergeometric function
    Defining formula
    R - a ⁡ ( b 1 , … , b n ; z 1 , … , z n ) Carlson-integral-R 𝑎 subscript 𝑏 1 … subscript 𝑏 𝑛 subscript 𝑧 1 … subscript 𝑧 𝑛 {\displaystyle{\displaystyle R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}% };\NVar{z_{1}},\dots,\NVar{z_{n}}\right)}}
    xml-id
    C19.S16.E9.m2andec
    0 references
    beta function
    Defining formula
    B ⁡ ( a , b ) Euler-Beta 𝑎 𝑏 {\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\right)}}
    xml-id
    C5.S12.E1.m2acdec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1agdec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3agdec
    0 references
    Q11820
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C19.S1.XMD3.m1ddec
    0 references
     
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