## G-Function

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

 (1)

where is the Euler-Mascheroni Constant. This is an Analytic Continuation of the function defined in the construction of the Glaisher-Kinkelin Constant
 (2)

which has the special values
 (3)

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

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

 (4)

 (5)

 (6)

 (7)

and has the special values
 (8) (9)

where
 (10)

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

An unrelated pair of functions are denoted and 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 -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 .'' 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 .'' 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.