G-Function

Defined in Whittaker and Watson (1990, p. 264) and also called the Barnes G-Function.

$$G(z+1)\equiv (2\pi)^{z/2} e^{-[z(z+1)+\gamma z^2]/2}\prod_{n... ... \left[{\left({1+{z\over n}}\right)^n e^{-z+z^2/(2n)}}\right],$$ (1)

where $\gamma$ is the Euler-Mascheroni Constant. This is an Analytic Continuation of the $G$ function defined in the construction of the Glaisher-Kinkelin Constant

$$G(n+1)\equiv {(n!)^n\over K(n+1)},$$ (2)

which has the special values

$$G(n)=\cases{ 0 & if $n=0, -1, -2, \ldots$\cr 1 & if $n=1$\cr 0! 1! 2! \cdots (n-2)! & if $n=2, 3, 4, \ldots$\cr}$$ (3)

for Integer $n$. This function is what Sloane and Plouffe (1995) call the Superfactorial, and the first few values for $n=1$, 2, ... are 1, 1, 1, 2, 12, 288, 34560, 24883200, 125411328000, 5056584744960000, ... (Sloane's A000178).

The $G$-function is the reciprocal of the Double Gamma Function. It satisfies

$$G(z+1)=\Gamma(z)G(z)$$ (4)

$${(n!)^n\over G(n+1)} = 1^1\cdot 2^2\cdot 3^3\cdots n^n$$ (5)

$${G'(z+1)\over G(z+1)} = {\textstyle{1\over 2}}\ln(2\pi)-{\textstyle{1\over 2}}-z+z{\Gamma'(z)\over \Gamma(z)}$$ (6)

$$\ln\left[{G(1-z)\over G(1+z)}\right]= \int_0^z \pi z\cot(\pi z)\,dz -z\ln(2\pi)$$ (7)

and has the special values

$$G({\textstyle{1\over 2}}) = A^{-3/2}\pi^{-1/4}e^{1/8} 2^{1/24}$$ (8)

$$G(1) = 1,$$ (9)

where

$$A=\mathop{\rm exp}\nolimits [{\textstyle{1\over 12}}-\zeta'(-1)] = 1.28242713\ldots.$$ (10)

The $G$-function can arise in spectral functions in mathematical physics (Voros 1987).

An unrelated pair of functions are denoted $g_n$ and $G_n$ and are known as Ramanujan g- and G-Functions.

See also Euler-Mascheroni Constant, Glaisher-Kinkelin Constant, K-Function, Meijer's G-Function, Ramanujan g- and G-Functions, Superfactorial

References

Barnes, E. W. ``The Theory of the $G$-Function.'' Quart. J. Pure Appl. Math. 31, 264-314, 1900.

Glaisher, J. W. L. ``On a Numerical Continued Product.'' Messenger Math. 6, 71-76, 1877.

Glaisher, J. W. L. ``On the Product $1^12^23^3\cdots n^n$.'' Messenger Math. 7, 43-47, 1878.

Glaisher, J. W. L. ``On Certain Numerical Products.'' Messenger Math. 23, 145-175, 1893.

Glaisher, J. W. L. ``On the Constant which Occurs in the Formula for $1^1 2^2 3^3\cdots\d{}n^n$.'' Messenger Math. 24, 1-16, 1894.

Kinkelin. ``Über eine mit der Gammafunktion verwandte Transcendente und deren Anwendung auf die Integralrechnung.'' J. Reine Angew. Math. 57, 122-158, 1860.

Sloane, N. J. A. Sequence A000178/M2049 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Voros, A. ``Spectral Functions, Special Functions and the Selberg Zeta Function.'' Commun. Math. Phys. 110, 439-465, 1987.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.