5 Gamma Function Properties 5.16 Sums 5.18

-Beta Functions

§5.17 Barnes’ -Function (Double Gamma Function)

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Keywords:: Barnes’ -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

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

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

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Defines:: $G\left(\NVar{z}\right)$ : Barnes’ -function (or double gamma function)
Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function and : 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 $G\left(n\right)=(n-2)!(n-3)!\cdots 1!,$ $n=2,3,\dots$ .

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Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ -function (or double gamma function), : factorial (as in ) and : 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 $G\left(z+1\right)=(2\pi)^{z/2}\exp\left(-\tfrac{1}{2}z(z+1)-\tfrac{1}{2}\gamma z% ^{2}\right)\*\prod_{k=1}^{\infty}\left(\left(1+\frac{z}{k}\right)^{k}\exp\left% (-z+\frac{z^{2}}{2k}\right)\right).$

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Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ -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, : nonnegative integer and : 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 $\operatorname{Ln}G\left(z+1\right)=\tfrac{1}{2}z\ln\left(2\pi\right)-\tfrac{1}% {2}z(z+1)+z\operatorname{Ln}\Gamma\left(z+1\right)-\int_{0}^{z}\operatorname{% Ln}\Gamma\left(t+1\right)\,\mathrm{d}t.$

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Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ -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 , $\int$ : integral, $\operatorname{Ln}\NVar{z}$ : general logarithm function, $\ln\NVar{z}$ : principal branch of logarithm function and : 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 $\operatorname{Ln}G\left(z+1\right)\sim\tfrac{1}{4}z^{2}+z\operatorname{Ln}% \Gamma\left(z+1\right)-\left(\tfrac{1}{2}z(z+1)+\tfrac{1}{12}\right)\ln z-\ln A% +\sum_{k=1}^{\infty}\frac{B_{2k+2}}{2k(2k+1)(2k+2)z^{2k}}.$

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Symbols:: $G\left(\NVar{z}\right)$ : Barnes’ -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, : nonnegative integer, : complex variable and : 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 is Glaisher’s constant, given by

5.17.6 $A=e^{C}=1.28242\;71291\;00622\;63687\;\ldots,$

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Symbols:: $\mathrm{e}$ : base of natural logarithm, : Glaisher’s constant and : 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 $C=\lim_{n\to\infty}\left(\sum_{k=1}^{n}k\ln k-\left(\tfrac{1}{2}n^{2}+\tfrac{1% }{2}n+\tfrac{1}{12}\right)\ln n+\tfrac{1}{4}n^{2}\right)=\frac{\gamma+\ln\left% (2\pi\right)}{12}-\frac{\zeta'\left(2\right)}{2\pi^{2}}=\frac{1}{12}-\zeta'% \left(-1\right),$

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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, : nonnegative integer, : nonnegative integer and : 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

-Gamma and

-Beta Functions

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