DLMF:15.4.E25 (Q5009)

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DLMF:15.4.E25
No description defined

    Statements

    Defining formula
    ∑ n = - ∞ ∞ Γ ⁡ ( a + n ) ⁢ Γ ⁡ ( b + n ) Γ ⁡ ( c + n ) ⁢ Γ ⁡ ( d + n ) = π 2 sin ⁡ ( π ⁢ a ) ⁢ sin ⁡ ( π ⁢ b ) ⁢ Γ ⁡ ( c + d - a - b - 1 ) Γ ⁡ ( c - a ) ⁢ Γ ⁡ ( d - a ) ⁢ Γ ⁡ ( c - b ) ⁢ Γ ⁡ ( d - b ) . superscript subscript 𝑛 Euler-Gamma 𝑎 𝑛 Euler-Gamma 𝑏 𝑛 Euler-Gamma 𝑐 𝑛 Euler-Gamma 𝑑 𝑛 superscript 𝜋 2 𝜋 𝑎 𝜋 𝑏 Euler-Gamma 𝑐 𝑑 𝑎 𝑏 1 Euler-Gamma 𝑐 𝑎 Euler-Gamma 𝑑 𝑎 Euler-Gamma 𝑐 𝑏 Euler-Gamma 𝑑 𝑏 {\displaystyle{\displaystyle\sum_{n=-\infty}^{\infty}\frac{\Gamma\left(a+n% \right)\Gamma\left(b+n\right)}{\Gamma\left(c+n\right)\Gamma\left(d+n\right)}=% \frac{\pi^{2}}{\sin\left(\pi a\right)\sin\left(\pi b\right)}\*\frac{\Gamma% \left(c+d-a-b-1\right)}{\Gamma\left(c-a\right)\Gamma\left(d-a\right)\Gamma% \left(c-b\right)\Gamma\left(d-b\right)}.}}
    0 references
    DLMF id
    DLMF:15.4.E25
    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.m2addec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2adec
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    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2addec
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    Q11650
    Defining formula
    n 𝑛 {\displaystyle{\displaystyle n}}
    xml-id
    C15.S1.XMD10.m1adec
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    Q11645
    Defining formula
    a 𝑎 {\displaystyle{\displaystyle a}}
    xml-id
    C15.S1.XMD4.m1rdec
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    Q11646
    Defining formula
    b 𝑏 {\displaystyle{\displaystyle b}}
    xml-id
    C15.S1.XMD5.m1gdec
    0 references
    Q11647
    Defining formula
    c 𝑐 {\displaystyle{\displaystyle c}}
    xml-id
    C15.S1.XMD6.m1ddec
    0 references
     
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