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

5.17.1		$G\left(z+1\right)$	$=\mathrm{\Gamma }\left(z\right)G\left(z\right),$
		$G\left(1\right)$	$=1,$
ⓘ Defines: $G\left(z\right)$: Barnes’ $G$-function (or double gamma function) Symbols: $\mathrm{\Gamma }\left(z\right)$: 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 Ch.5

5.17.2		$$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$
$n=2,3,\mathrm{\dots }$.
ⓘ Symbols: $G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer Permalink: http://dlmf.nist.gov/5.17.E2 Encodings: TeX, pMML, png See also: Annotations for §5.17 and Ch.5

5.17.3		$$G\left(z+1\right)={(2\pi )}^{z/2}\exp \left(-\frac{1}{2}z(z+1)-\frac{1}{2}\gamma z^2\right)\prod _{k=1}^{\mathrm{\infty }}\left({\left(1+\frac{z}{k}\right)}^k\exp \left(-z+\frac{z^2}{2k}\right)\right).$$
ⓘ Symbols: $G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $k$: nonnegative integer and $z$: complex variable Permalink: http://dlmf.nist.gov/5.17.E3 Encodings: TeX, pMML, png See also: Annotations for §5.17 and Ch.5

5.17.4		$$\mathrm{Ln}G\left(z+1\right)=\frac{1}{2}z\ln \left(2\pi \right)-\frac{1}{2}z(z+1)+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-{\int }_0^z\mathrm{Ln}\mathrm{\Gamma }\left(t+1\right)dt.$$
ⓘ Symbols: $G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function and $z$: complex variable Permalink: http://dlmf.nist.gov/5.17.E4 Encodings: TeX, pMML, png See also: Annotations for §5.17 and Ch.5

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

When $z\to \mathrm{\infty }$ in ,

5.17.5		$$\mathrm{Ln}G\left(z+1\right)\sim \begin{array}{l}\frac{1}{4}{z}^2+z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)-\left(\frac{1}{2}z(z+1)+\frac{1}{12}\right)\mathrm{Ln}z-\ln A\\ +\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.\end{array}$$
ⓘ Symbols: $G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer, $z$: complex variable and $A$: Glaisher’s constant Referenced by: §5.17, §5.17, Erratum (V1.0.7) for 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 $z\mathrm{Ln}\mathrm{\Gamma }\left(z+1\right)$ was incorrectly stated as $z\mathrm{\Gamma }\left(z+1\right)$. (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 Ch.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		$$A={\mathrm{e}}^C=\mathrm{1.28242\hspace{0.33em}71291\hspace{0.33em}00622\hspace{0.33em}63687}\mathrm{\dots },$$
ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $A$: 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 Ch.5

where

5.17.7		$$C=\underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^{n}k\ln k-\left(\frac{1}{2}{n}^2+\frac{1}{2}n+\frac{1}{12}\right)\ln n+\frac{1}{4}{n}^2\right)=\frac{\gamma +\ln \left(2\pi \right)}{12}-\frac{{\zeta }^{\prime }\left(2\right)}{2{\pi }^2}=\frac{1}{12}-{\zeta }^{\prime }\left(-1\right),$$
ⓘ Symbols: $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer, $k$: nonnegative integer and $C$: log of Glaisher’s Constant Referenced by: §5.17 Permalink: http://dlmf.nist.gov/5.17.E7 Encodings: TeX, pMML, png See also: Annotations for §5.17 and Ch.5

and ${\zeta }^{\prime }$ 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).