Barnes G-function

In mathematics, the Barnes G-function (typically denoted G(z)) is a function that is an extension of superfactorials to the complex numbers. It is related to the Gamma function, the K-function and the Glaisher-Kinkelin constant, and was named after mathematician Ernest William Barnes.

Formally, the Barnes G-function is defined (in the form of a Weierstrass product) as

$G(z+1)=(2\pi)^{z/2} e^{-[z(z+1)+\gamma z^2]/2}\prod_{n=1}^\infty \left[\left(1+\frac{z}{n}\right)^ne^{-z+z^2/(2n)}\right]$

[edit] Functional equation and special values

The Barnes G-function satisfies the functional equation

G (z + 1) = Γ(z) G (z)

with normalisation G(1)=1. The functional equation implies that G takes the following values at integer arguments:

$G(n)=\begin{cases} 0&\mbox{if }n=0,-1,-2,\dots\\ \prod_{i=0}^{n-2} i!&\mbox{if }n=1,2,\dots\end{cases}$

and thus

$G(n)=\frac{(\Gamma(n))^{n-1}}{K(n)}$

where Γ denotes the Gamma function and K denotes the K-function.

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