DLMF:20.11.E2 (Q6843)

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DLMF:20.11.E2
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    1 n G ( m , n ) = 1 n k = 0 n - 1 e - π i k 2 m / n = e - π i / 4 m j = 0 m - 1 e π i j 2 n / m = e - π i / 4 m G ( - n , m ) . 1 𝑛 𝐺 𝑚 𝑛 1 𝑛 superscript subscript 𝑘 0 𝑛 1 superscript 𝑒 𝜋 𝑖 superscript 𝑘 2 𝑚 𝑛 superscript 𝑒 𝜋 𝑖 4 𝑚 superscript subscript 𝑗 0 𝑚 1 superscript 𝑒 𝜋 𝑖 superscript 𝑗 2 𝑛 𝑚 superscript 𝑒 𝜋 𝑖 4 𝑚 𝐺 𝑛 𝑚 {\displaystyle{\displaystyle\frac{1}{\sqrt{n}}G(m,n)=\frac{1}{\sqrt{n}}\sum% \limits_{k=0}^{n-1}e^{-\pi ik^{2}m/n}=\frac{e^{-\pi i/4}}{\sqrt{m}}\sum\limits% _{j=0}^{m-1}e^{\pi ij^{2}n/m}=\frac{e^{-\pi i/4}}{\sqrt{m}}G(-n,m).}}
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    DLMF:20.11.E2
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