# §5.17 Barnes’ $G$-Function (Double Gamma Function)

 5.17.1 $\displaystyle G\left(z+1\right)$ $\displaystyle=\Gamma\left(z\right)G\left(z\right),$ $\displaystyle G\left(1\right)$ $\displaystyle=1,$ ⓘ Defines: $G\left(\NVar{z}\right)$: Barnes’ $G$-function (or double gamma function) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $z$: complex variable Permalink: http://dlmf.nist.gov/5.17.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 5.17 and 5
 5.17.2 $G\left(n\right)=(n-2)!(n-3)!\cdots 1!,$ $n=2,3,\dots$.
 5.17.3 $G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$
 5.17.4 $\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\mathrm{d}t.$

In this equation (and in (5.17.5) below), the $\operatorname{Ln}$’s have their principal values on the positive real axis and are continued via continuity, as in §4.2(i).

When $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta\;(<\pi)$,

 5.17.5 $\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)% \operatorname{Ln}z-\ln A+\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2% k}}.$ ⓘ Symbols: $G\left(\NVar{z}\right)$: Barnes’ $G$-function (or double gamma function), $B_{\NVar{n}}$: Bernoulli numbers, $\Gamma\left(\NVar{z}\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\operatorname{Ln}\NVar{z}$: general logarithm function, $\ln\NVar{z}$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant Referenced by: §5.17, §5.17, Equation (5.17.5) Permalink: http://dlmf.nist.gov/5.17.E5 Encodings: TeX, pMML, png Errata (effective with 1.0.7): Originally the term $z\operatorname{Ln}\Gamma\left(z+1\right)$ was incorrectly stated as $z\Gamma\left(z+1\right)$. (This error was reported subsequently by Nick Jones on December 11, 2013.) Reported 2013-08-01 by Gergő Nemes See also: Annotations for 5.17 and 5

For error bounds and an exponentially-improved extension, see Nemes (2014). Here $B_{2k+2}$ is the Bernoulli number (§24.2(i)), and $A$ is Glaisher’s constant, given by

 5.17.6 $A=e^{C}=1.28242\;71291\;00622\;63687\;\ldots,$ ⓘ Symbols: $\mathrm{e}$: base of exponential function, $A$: Glaisher’s constant and $C$: log of Glaisher’s Constant Notes: For more digits see OEIS Sequence A074962; see also Sloane (2003). Permalink: http://dlmf.nist.gov/5.17.E6 Encodings: TeX, pMML, png See also: Annotations for 5.17 and 5

where

 5.17.7 $C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1% }{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\gamma+\ln\left% (2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'% \left(-1\right),$

and $\zeta'$ is the derivative of the zeta function (Chapter 25).

For Glaisher’s constant see also Greene and Knuth (1982, p. 100) and §2.10(i).