§ 5.17. Barnes' $G$ -Function (Double Gamma Function)

Notes:: See Whittaker and Watson (1927, p. 264). For (5.17.7) see Olver (1997b, p. 292) and the differentiated form of (Ch.25).
Keywords:: Barnes' $G$ -function, Glaisher's constant
Referenced by:: §2.10(i)
Permalink:: http://dlmf.nist.gov/5.17

5.17.1

$G\!\left(z+1\right)=\Gamma\!\left(z\right)G\!\left(z\right),$

$G\!\left(1\right)=1,$

Defines:: $G\!\left(z\right)$ : Barnes $G$ -function
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E1
Encodings:: TeX, TeX, pMathML, pMathML, png, png

5.17.2 $G\!\left(n\right)=(n-2)!(n-3)!\cdots 1!,$ $n=2,3,\dots$ .

Symbols:: $G\!\left(z\right)$ : Barnes $G$ -function and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.17.E2
Encodings:: TeX, pMathML, png

5.17.3 $G\!\left(z+1\right)=(2\pi)^{{z/2}}\exp\!\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\EulerConstant z^{2}\right)\times\prod _{{k=1}}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\!\left(-z+\frac{z^{2}}{2k}\right)\right).$

Symbols:: $G\!\left(z\right)$ : Barnes $G$ -function, $\EulerConstant$ : Euler's constant, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E3
Encodings:: TeX, pMathML, png

5.17.4 $\mathrm{Ln}G\!\left(z+1\right)=\tfrac{1}{2}z\ln\!\left(2\pi\right)-\tfrac{1}{2}z(z+1)+z\mathrm{Ln}\Gamma\!\left(z+1\right)-\int _{0}^{z}\mathrm{Ln}\Gamma\!\left(t+1\right)dt.$

Symbols:: $G\!\left(z\right)$ : Barnes $G$ -function, $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E4
Encodings:: TeX, pMathML, png

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 §Ch.4.

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

5.17.5 $\mathrm{Ln}G\!\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\Gamma\!\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\mathrm{Ln}z-\ln A+\sum _{{k=1}}^{\infty}\frac{B_{{2k+2}}}{2k(2k+1)(2k+2)z^{{2k}}};$

Defines:: $A$ : Glaisher's constant
Symbols:: $G\!\left(z\right)$ : Barnes $G$ -function, $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §5.17
Permalink:: http://dlmf.nist.gov/5.17.E5
Encodings:: TeX, pMathML, png

see Ferreira and López (2001). This reference also provides bounds for the error term. Here $B_{{2k+2}}$ is the Bernoulli number (§Ch.24), and $A$ is Glaisher's constant, given by

5.17.6 $A=e^{C}=1.28242\; 71291\; 0 0 622\; 63687\;\ldots,$

Notes:: For more digits see OEIS Sequence A074962; see also Sloane (2003).
Defines:: $A$ : Glaisher's constant and $C$ : log of Glaisher's Constant
Permalink:: http://dlmf.nist.gov/5.17.E6
Encodings:: TeX, pMathML, png

where

5.17.7 $C=\lim _{{n\to\infty}}\left(\sum _{{k=1}}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1}{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\EulerConstant+\ln\!\left(2\pi\right)}{12}-\frac{{{\zeta}^{{\prime}}}\!\left(2\right)}{2\pi^{2}}=\frac{1}{12}-{{\zeta}^{{\prime}}}\!\left(-1\right),$

Defines:: $C$ : log of Glaisher's Constant
Symbols:: $\EulerConstant$ : Euler's constant, $n$ : nonnegative integer and $k$ : nonnegative integer
Referenced by:: §5.17
Permalink:: http://dlmf.nist.gov/5.17.E7
Encodings:: TeX, pMathML, png

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).

§ 5.17. Barnes' -Function (Double Gamma Function)

§ 5.17. Barnes' $G$ -Function (Double Gamma Function)