§5.17 Barnes’ $\mathop{G\/}\nolimits$ -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 (25.4.1).
Keywords:: Barnes’ $G$ -function, Glaisher’s constant
Referenced by:: ¶ ‣ §2.10(i)
Permalink:: http://dlmf.nist.gov/5.17

5.17.1

$\mathop{G\/}\nolimits\!\left(z+1\right)=\mathop{\Gamma\/}\nolimits\!\left(z\right)\mathop{G\/}\nolimits\!\left(z\right),$

$\mathop{G\/}\nolimits\!\left(1\right)=1,$

Defines:: $\mathop{G\/}\nolimits\!\left(z\right)$ : Barnes’ $G$ -function (or double gamma function)
Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

5.17.2 $\mathop{G\/}\nolimits\!\left(n\right)=(n-2)!(n-3)!\cdots 1!,$ $n=2,3,\dots$ .

Symbols:: $\mathop{G\/}\nolimits\!\left(z\right)$ : Barnes’ $G$ -function (or double gamma function), $!$ : $n!$ : factorial and $n$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/5.17.E2
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{G\/}\nolimits\!\left(z\right)$ : Barnes’ $G$ -function (or double gamma function), $\EulerConstant$ : Euler’s constant, $\mathop{\exp\/}\nolimits z$ : exponential function, $k$ : nonnegative integer and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E3
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{G\/}\nolimits\!\left(z\right)$ : Barnes’ $G$ -function (or double gamma function), $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/5.17.E4
Encodings:: TeX, pMML, png

In this equation (and in (5.17.5) below), the $\mathop{\mathrm{Ln}\/}\nolimits$ ’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 $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta\;(<\pi)$ ,

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

Symbols:: $\mathop{G\/}\nolimits\!\left(z\right)$ : Barnes’ $G$ -function (or double gamma function), $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $A$ : Glaisher’s constant, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\sim$ : asymptotic equality, $k$ : nonnegative integer and $z$ : complex variable
Referenced by:: §5.17
Permalink:: http://dlmf.nist.gov/5.17.E5
Encodings:: TeX, pMML, png

see Ferreira and López (2001). This reference also provides bounds for the error term. Here $\mathop{B_{{2k+2}}\/}\nolimits$ is the Bernoulli number (§24.2(i)), 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
Symbols:: $e$ : base of exponential function and $C$ : log of Glaisher’s constant
Permalink:: http://dlmf.nist.gov/5.17.E6
Encodings:: TeX, pMML, png

where

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

Defines:: $C$ : log of Glaisher’s constant
Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $n$ : nonnegative integer and $k$ : nonnegative integer
Referenced by:: §5.17
Permalink:: http://dlmf.nist.gov/5.17.E7
Encodings:: TeX, pMML, png

and ${\mathop{\zeta\/}\nolimits^{{\prime}}}$ 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’ $\mathop{G\/}\nolimits$ -Function (Double Gamma Function)