DLMF:5.14.E5 (Q2167)

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DLMF:5.14.E5
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    Statements

    Defining formula
    ∫ [ 0 , ∞ ) n t 1 ⁢ t 2 ⁢ ⋯ ⁢ t m ⁢ | Δ ⁢ ( t 1 , … , t n ) | 2 ⁢ c ⁢ ∏ k = 1 n t k a - 1 ⁢ e - t k ⁢ d t k = ∏ k = 1 m ( a + ( n - k ) ⁢ c ) ⁢ ∏ k = 1 n Γ ⁡ ( a + ( n - k ) ⁢ c ) ⁢ Γ ⁡ ( 1 + k ⁢ c ) ( Γ ⁡ ( 1 + c ) ) n , subscript superscript 0 𝑛 subscript 𝑡 1 subscript 𝑡 2 ⋯ subscript 𝑡 𝑚 superscript Δ subscript 𝑡 1 … subscript 𝑡 𝑛 2 𝑐 superscript subscript product 𝑘 1 𝑛 superscript subscript 𝑡 𝑘 𝑎 1 superscript 𝑒 subscript 𝑡 𝑘 subscript 𝑡 𝑘 superscript subscript product 𝑘 1 𝑚 𝑎 𝑛 𝑘 𝑐 superscript subscript product 𝑘 1 𝑛 Euler-Gamma 𝑎 𝑛 𝑘 𝑐 Euler-Gamma 1 𝑘 𝑐 superscript Euler-Gamma 1 𝑐 𝑛 {\displaystyle{\displaystyle\int_{[0,\infty)^{n}}t_{1}t_{2}\cdots t_{m}|\Delta% (t_{1},\dots,t_{n})|^{2c}\prod_{k=1}^{n}t_{k}^{a-1}e^{-t_{k}}\mathrm{d}t_{k}=% \prod_{k=1}^{m}(a+(n-k)c)\frac{\prod_{k=1}^{n}\Gamma\left(a+(n-k)c\right)% \Gamma\left(1+kc\right)}{(\Gamma\left(1+c\right))^{n}},}}
    0 references
    DLMF id
    DLMF:5.14.E5
    0 references
    Symbols used
    gamma function
    Defining formula
    Γ ⁡ ( z ) Euler-Gamma 𝑧 {\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
    xml-id
    C5.S2.E1.m2acdec
    0 references
    Q11243
    Defining formula
    [ a , b ) 𝑎 𝑏 {\displaystyle{\displaystyle[\NVar{a},\NVar{b})}}
    xml-id
    introduction.Sx4.p2.t1.r1.m4adec
    0 references
    Q10770
    Defining formula
    d x 𝑥 {\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
    xml-id
    C1.S4.SS4.m1acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2adec
    0 references
    Q10771
    Defining formula
    ∫ {\displaystyle{\displaystyle\int}}
    xml-id
    C1.S4.SS4.m3acdec
    0 references
    Q11276
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C5.S1.XMD2.m1adec
    0 references
    Q11274
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C5.S1.XMD3.m1ddec
    0 references
    Q11279
    Defining formula
    k 𝑘 {\displaystyle{\displaystyle k}}
    xml-id
    C5.S1.XMD4.m1cdec
    0 references
    Q11275
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C5.S1.XMD8.m1adec
    0 references
    product (locally)
    Defining formula
    Δ Δ {\displaystyle{\displaystyle\Delta}}
    xml-id
    C5.S14.XMD2.m1bdec
    0 references
     
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