Multiple gamma function

Summary

The multiple gamma functions $\Gamma_N(w \mid a_1, \ldots, a_N)$ of E. W. Barnes generalise the gamma function to $N$ periods. They are built from the Barnes zeta function $\sum (w + n_1 a_1 + \cdots + n_N a_N)^{-s}$ by zeta regularisation, as the gamma function is built from the Hurwitz zeta function. Shifting $w$ by a period $a_j$ divides $\Gamma_N$ by a multiple gamma function of order $N - 1$. The Barnes G-function is essentially the case $N = 2$ with $a_1 = a_2 = 1$.