DLMF:35.7.E2 (Q10505)

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DLMF:35.7.E2
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    Statements

    DLMF defining formula
    P ν ( γ , δ ) ⁡ ( 𝐓 ) = Γ m ⁡ ( γ + ν + 1 2 ⁢ ( m + 1 ) ) Γ m ⁡ ( γ + 1 2 ⁢ ( m + 1 ) ) ⁢ F 1 2 ⁡ ( - ν , γ + δ + ν + 1 2 ⁢ ( m + 1 ) γ + 1 2 ⁢ ( m + 1 ) ; 𝐓 ) , Jacobi-function-of-matrix-P 𝛾 𝛿 𝜈 𝐓 multivariate-Euler-Gamma 𝑚 𝛾 𝜈 1 2 𝑚 1 multivariate-Euler-Gamma 𝑚 𝛾 1 2 𝑚 1 Gauss-hypergeometric-of-matrix-pFq 2 1 𝜈 𝛾 𝛿 𝜈 1 2 𝑚 1 𝛾 1 2 𝑚 1 𝐓 {\displaystyle{\displaystyle P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=% \frac{\Gamma_{m}\left(\gamma+\nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(% \gamma+\frac{1}{2}(m+1)\right)}\*{{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+% \frac{1}{2}(m+1)\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),}}
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    constraint
    γ , δ , ν ∈ ℂ 𝛾 𝛿 𝜈 {\displaystyle{\displaystyle\gamma,\delta,\nu\in\mathbb{C}}}
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    ℜ ⁡ ( γ ) > - 1 𝛾 1 {\displaystyle{\displaystyle\Re(\gamma)>-1}}
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    𝟎 < 𝐓 < 𝐈 0 𝐓 𝐈 {\displaystyle{\displaystyle\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}}}
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    γ , δ , ν ∈ ℂ 𝛾 𝛿 𝜈 ℂ {\displaystyle{\displaystyle\gamma,\delta,\nu\in\mathbb{C}}}
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    ℜ ⁡ ( γ ) > - 1 𝛾 1 {\displaystyle{\displaystyle\Re(\gamma)>-1}}
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    Symbols used
    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.m2aadec
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    Q10897
    DLMF defining formula
    ℂ {\displaystyle{\displaystyle\mathbb{C}}}
    xml-id
    introduction.Sx4.p1.t1.r1.m2adec
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    Q10758
    DLMF defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2adec
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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.m1adec
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    real part
    DLMF defining formula
    ℜ ⁡ absent {\displaystyle{\displaystyle\Re}}
    xml-id
    C1.S9.E2.m1adec
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    space of real symmetric matrices
    DLMF defining formula
    𝐓 𝐓 {\displaystyle{\displaystyle\mathbf{T}}}
    xml-id
    C35.S1.XMD11.m1adec
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    positive integer
    DLMF defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C35.S1.XMD8.m1adec
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    Identifiers

     
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