Contents

# Contents

## Idea

Much like the Gamma function generalizes the functional equation

$\Gamma(z + 1) \;=\; z \, \Gamma(z)$

to non-integer values of $z$, so the Barnes $G$-function $G(-)$ corresponds to the functional equation

(1)$G(z + 1) \;=\; \Gamma(z) \, G(z) \,.$

Just as for $z =n \in \mathbb{N}$, $\Gamma(n+1) = n!$, so $G(n+2) = n!(n-1)! \cdots 1!$.

(…)

## Properties

### Special values

(2)$G(1) \;=\; 1$
$G(1/2) \;=\; 2^{1/24} \cdot e^{ \tfrac{3}{2} \zeta^'(-1) } \cdot \pi^{ - 1/4 } \,.$

(WP here)

### Stirling-like asymptotic expansion

$ln G(1 + z) \;=\; z^2 \left( \tfrac{1}{2} ln(z) - \tfrac{3}{4} \right) + \tfrac{1}{2} ln(2 \pi) z - \tfrac{1}{12} ln(z) + \zeta^'(-1) + \mathcal{O}(z^{-1}) \,,$

where $\zeta$ denotes the Riemann zeta function.

(e.g. WP here, WMW (14))

### Relations to the Gamma-function

A version of the Gauss multiplication formula for the Gamma function:

###### Proposition

$\underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \frac { G(N/2+ 1) \cdot G(N/2 + 1/2) } { G(1/2) } \,.$

(Kotěšovec 13, p. 2)
###### Proof

By repeated use of the translation formula (1) and using the initial value $G(1) = 1$ (2):

In the case that $N = 2M$ in an even number:

$\underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \underoverset {j = 1} {M} {\prod} \Gamma(j) \cdot \underoverset {j = 1} {M} {\prod} \Gamma(j - \frac{1}{2}) \;=\; G(M+1) \cdot G(M +1/2) /G(1/2) \;=\; G(N/2 + 1) \cdot G(N/2 + 1/2)/G(1/2).$

In the case that $N = 2M+1$ is an odd number:

$\underoverset {j = 1} {N} {\prod} \Gamma(j/2) \;=\; \underoverset {j = 1} {M} {\prod} \Gamma(j) \cdot \underoverset {j = 1} {M+1} {\prod} \Gamma(j - \frac{1}{2}) \;=\; G(M+1) \cdot G(M +3/2) /G(1/2) \;=\; G(N/2 + 1/2) \cdot G(N/2 + 1)/G(1/2).$