Barnes G-function - Multiplication Formula

Multiplication Formula

Like the Gamma function, the G-function also has a multiplication formula:

$G(nz)= K(n) n^{n^{2}z^{2}/2-nz} (2\pi)^{-\frac{n^2-n}{2}z}\prod_{i=0}^{n-1}\prod_{j=0}^{n-1}G\left(z+\frac{i+j}{n}\right)$

where is a constant given by:

$K(n)= e^{-(n^2-1)\zeta^\prime(-1)} \cdot n^{\frac{5}{12}}\cdot(2\pi)^{(n-1)/2}\,=\, (Ae^{-\frac{1}{12}})^{n^2-1}\cdot n^{\frac{5}{12}}\cdot (2\pi)^{(n-1)/2}.$

Here is the derivative of the Riemann zeta function and is the Glaisher–Kinkelin constant.

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