Barnes G-function

Summary

The Barnes G-function is an entire function $G$ with $G(1) = 1$ and $G(z+1) = \Gamma(z)\, G(z)$, where $\Gamma$ is the gamma function. It plays for the gamma function the role that the gamma function plays for the identity function. At positive integers it gives the superfactorials $G(n) = 0!\,1!\cdots(n-2)!$. It is, up to an elementary factor, a multiple gamma function of order two, and it appears in random matrix theory and in determinant formulas.