DLMF:35.8.E7 (Q10521): Difference between revisions

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DLMF:35.8.E7
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    Statements

    DLMF defining formula
    F 2 3 ⁡ ( a 1 , a 2 , a 3 b 1 , b 2 ; 𝐈 ) = Γ m ⁡ ( b 1 ) ⁢ Γ m ⁡ ( b 2 ) ⁢ Γ ⁡ ( c ) Γ m ⁡ ( a 1 ) ⁢ Γ m ⁡ ( c + a 2 ) ⁢ Γ ⁡ ( c + a 3 ) ⁢ F 2 3 ⁡ ( b 1 - a 1 , b 2 - a 2 , c c + a 2 , c + a 3 ; 𝐈 ) , Gauss-hypergeometric-of-matrix-pFq 3 2 subscript 𝑎 1 subscript 𝑎 2 subscript 𝑎 3 subscript 𝑏 1 subscript 𝑏 2 𝐈 multivariate-Euler-Gamma 𝑚 subscript 𝑏 1 multivariate-Euler-Gamma 𝑚 subscript 𝑏 2 Euler-Gamma 𝑐 multivariate-Euler-Gamma 𝑚 subscript 𝑎 1 multivariate-Euler-Gamma 𝑚 𝑐 subscript 𝑎 2 Euler-Gamma 𝑐 subscript 𝑎 3 Gauss-hypergeometric-of-matrix-pFq 3 2 subscript 𝑏 1 subscript 𝑎 1 subscript 𝑏 2 subscript 𝑎 2 𝑐 𝑐 subscript 𝑎 2 𝑐 subscript 𝑎 3 𝐈 {\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b% _{2}};\mathbf{I}\right)=\frac{\Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{2% }\right)\Gamma\left(c\right)}{\Gamma_{m}\left(a_{1}\right)\Gamma_{m}\left(c+a_% {2}\right)\Gamma\left(c+a_{3}\right)}\*{{}_{3}F_{2}}\left({b_{1}-a_{1},b_{2}-a% _{2},c\atop c+a_{2},c+a_{3}};\mathbf{I}\right),}}
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    constraint
    ℜ ⁡ ( b 1 ) subscript 𝑏 1 {\displaystyle{\displaystyle\Re(b_{1})}}
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    ℜ ⁡ ( b 2 ) subscript 𝑏 2 {\displaystyle{\displaystyle\Re(b_{2})}}
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    ℜ ⁡ ( c ) > 1 2 ⁢ ( m - 1 ) 𝑐 1 2 𝑚 1 {\displaystyle{\displaystyle\Re(c)>\frac{1}{2}(m-1)}}
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    Symbols used
    gamma function
    DLMF defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2adec
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    generalized hypergeometric function of matrix argument
    DLMF defining formula
    F q p ⁡ ( a 1 , … , a p ; b 1 , … , b q ; 𝐓 ) Gauss-hypergeometric-of-matrix-pFq 𝑝 𝑞 subscript 𝑎 1 … subscript 𝑎 𝑝 subscript 𝑏 1 … subscript 𝑏 𝑞 𝐓 {\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots% ,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{\mathbf{T}}\right)}}
    xml-id
    C35.S8.E1.m2afdec
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    Q12728
    DLMF defining formula
    Γ m ⁡ ( a ) multivariate-Euler-Gamma 𝑚 𝑎 {\displaystyle{\displaystyle\Gamma_{\NVar{m}}\left(\NVar{a}\right)}}
    xml-id
    C35.S3.SS1.m1acdec
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    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1aadec
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    complex variable
    DLMF defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C35.S1.XMD1.m1ddec
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    complex variable
    DLMF defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C35.S1.XMD2.m1ddec
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    complex variable
    DLMF defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C35.S1.XMD3.m1adec
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    positive integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1cdec
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    Identifiers

     
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