DLMF:21.6.E6 (Q6894)

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DLMF:21.6.E6
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    Statements

    Defining formula
    θ ⁡ ( 𝐱 + 𝐲 + 𝐮 + 𝐯 2 | 𝛀 ) ⁢ θ ⁡ ( 𝐱 + 𝐲 - 𝐮 - 𝐯 2 | 𝛀 ) ⁢ θ ⁡ ( 𝐱 - 𝐲 + 𝐮 - 𝐯 2 | 𝛀 ) ⁢ θ ⁡ ( 𝐱 - 𝐲 - 𝐮 + 𝐯 2 | 𝛀 ) = 1 2 g ⁢ ∑ 𝜶 ∈ 1 2 ⁢ ℤ g / ℤ g ∑ 𝜷 ∈ 1 2 ⁢ ℤ g / ℤ g e 2 ⁢ π ⁢ i ⁢ ( 2 ⁢ 𝜶 ⋅ 𝛀 ⋅ 𝜶 + 𝜶 ⋅ [ 𝐱 + 𝐲 + 𝐮 + 𝐯 ] ) ⁢ θ ⁡ ( 𝐱 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) ⁢ θ ⁡ ( 𝐲 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) ⁢ θ ⁡ ( 𝐮 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) ⁢ θ ⁡ ( 𝐯 + 𝛀 ⁢ 𝜶 + 𝜷 | 𝛀 ) , Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 Riemann-theta 𝐱 𝐲 𝐮 𝐯 2 𝛀 1 superscript 2 𝑔 subscript 𝜶 1 2 𝑔 𝑔 subscript 𝜷 1 2 𝑔 𝑔 superscript 𝑒 2 𝜋 𝑖 ⋅ 2 𝜶 𝛀 𝜶 ⋅ 𝜶 delimited-[] 𝐱 𝐲 𝐮 𝐯 Riemann-theta 𝐱 𝛀 𝜶 𝜷 𝛀 Riemann-theta 𝐲 𝛀 𝜶 𝜷 𝛀 Riemann-theta 𝐮 𝛀 𝜶 𝜷 𝛀 Riemann-theta 𝐯 𝛀 𝜶 𝜷 𝛀 {\displaystyle{\displaystyle\theta\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}% +\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x% }+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)% \theta\left(\frac{\mathbf{x}-\mathbf{y}+\mathbf{u}-\mathbf{v}}{2}\middle|% \boldsymbol{{\Omega}}\right)\theta\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}% +\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)=\frac{1}{2^{g}}\sum_{% \boldsymbol{{\alpha}}\in\frac{1}{2}{\mathbb{Z}^{g}}/{\mathbb{Z}^{g}}}\,\sum_{% \boldsymbol{{\beta}}\in\frac{1}{2}{\mathbb{Z}^{g}}/{\mathbb{Z}^{g}}}e^{2\pi i% \left(2\boldsymbol{{\alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha% }}+\boldsymbol{{\alpha}}\cdot[\mathbf{x}+\mathbf{y}+\mathbf{u}+\mathbf{v}]% \right)}\*\theta\left(\mathbf{x}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+% \boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)\theta\left(\mathbf{y}% +\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|% \boldsymbol{{\Omega}}\right)\theta\left(\mathbf{u}+\boldsymbol{{\Omega}}% \boldsymbol{{\alpha}}+\boldsymbol{{\beta}}\middle|\boldsymbol{{\Omega}}\right)% \theta\left(\mathbf{v}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{% {\beta}}\middle|\boldsymbol{{\Omega}}\right),}}
    0 references
    DLMF id
    DLMF:21.6.E6
    0 references
    Symbols used
    Riemann theta function
    Defining formula
    θ ⁡ ( 𝐳 | 𝛀 ) Riemann-theta 𝐳 𝛀 {\displaystyle{\displaystyle\theta\left(\NVar{\mathbf{z}}\middle|\NVar{% \boldsymbol{{\Omega}}}\right)}}
    xml-id
    C21.S2.E1.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2abdec
    0 references
    Q10784
    Defining formula
    ∈ {\displaystyle{\displaystyle\in}}
    xml-id
    introduction.Sx4.p1.t1.r10.m2abdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2abdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2abdec
    0 references
    Q10818
    Defining formula
    ℤ {\displaystyle{\displaystyle\mathbb{Z}}}
    xml-id
    introduction.Sx4.p2.t1.r20.m2abdec
    0 references
    Q11959
    Defining formula
    g 𝑔 {\displaystyle{\displaystyle g}}
    xml-id
    C21.S1.XMD1.m1cdec
    0 references
    Q11960
    Defining formula
    𝛀 𝛀 {\displaystyle{\displaystyle\boldsymbol{{\Omega}}}}
    xml-id
    C21.S1.XMD3.m1bdec
    0 references
    Q11961
    Defining formula
    𝜶 𝜶 {\displaystyle{\displaystyle\boldsymbol{{\alpha}}}}
    xml-id
    C21.S1.XMD4.m1dec
    0 references
    Q11962
    Defining formula
    𝜷 𝜷 {\displaystyle{\displaystyle\boldsymbol{{\beta}}}}
    xml-id
    C21.S1.XMD5.m1dec
    0 references
     
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