nLab Barnes G-function

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 $G(n+2) = n!(n-1)! \cdots 1!$.

Definition

(…)

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:

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

References

See also:

• Wikipedia, Barnes G-function

• WolframMathWorld, Barnes G-Function

In the context of counting of standard Young tableaux of bounded height:

• Václav Kotěšovec, Asymptotic of Young tableaux of bounded height, 2013 (pdf, pdf)