5 Gamma Function Properties 5.16 Sums 5.18

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-Beta Functions

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

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Keywords:: Barnes’ $G$ -function, Glaisher’s constant, asymptotic expansion, definition, infinite product, integral representation, recurrence relation
Notes:: See Whittaker and Watson (1927, p. 264). For (5.17.7) see Olver (1997b, p. 292) and the differentiated form of (25.4.1).
Referenced by:: §2.10(i), Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/5.17
Changes and Additions (effective with 1.0.11):: The original reference to Ferreira and López (2001) that appeared after (5.17.5), and the original sentence that followed this reference, have been removed and replaced with a sentence referring to Nemes (2014a).
See also:: Annotations for Ch.5

5.17.1	$\displaystyle G\left(z+1\right)$	$\displaystyle=\Gamma\left(z\right)G\left(z\right),$
5.17.1	$\displaystyle G\left(1\right)$	$\displaystyle=1,$
ⓘ 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 Ch.5

5.17.2		$G\left(n\right)=(n-2)!(n-3)!\cdots 1!,$
$n=2,3,\dots$ .
ⓘ Symbols: $G\left(\NVar{z}\right)$ : 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		$G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$
ⓘ Symbols: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$ : 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		$\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\,\mathrm{d}t.$
ⓘ Symbols: $G\left(\NVar{z}\right)$ : Barnes’ $G$ -function (or double gamma function), $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : 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 $\operatorname{Ln}$ ’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		$\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$
ⓘ 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, Erratum (V1.0.7) for Equation (5.17.5), Erratum (V1.1.11) for Equation (5.17.5) Permalink: http://dlmf.nist.gov/5.17.E5 Encodings: TeX, pMML, png Errata (effective with 1.1.11): For consistency we have replaced $\operatorname{Ln}z$ by $\ln z$ . 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 Ch.5

For error bounds and an exponentially-improved extension, see Nemes (2014a). 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 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		$C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1% }{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\gamma+\ln\left% (2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'% \left(-1\right),$
ⓘ Symbols: $\gamma$ : Euler’s constant, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : 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 $\zeta'$ 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).

5.16 Sums 5.18

q

-Gamma and

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-Beta Functions

NIST

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

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