In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the Gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. Up to elementary factors, it is a special case of the double gamma function.
Formally, the Barnes G-function is defined (in the form of a Weierstrass product) as
Difference equation, functional equation and special values
The Barnes G-function satisfies the difference equation
with normalisation G(1) = 1. The difference equation implies that G takes the following values at integer arguments:
Like the Gamma function, the G-function also has a multiplication formula:
where is a constant given by:
The logarithm of G(z + 1) has the following asymptotic expansion, as established by Barnes:
Here the are the Bernoulli numbers and is the Glaisher–Kinkelin constant. (Note that somewhat confusingly at the time of Barnes  the Bernoulli number would have been written as , but this convention is no longer current.) This expansion is valid for in any sector not containing the negative real axis with large.
- E.W.Barnes, "The theory of the G-function", Quarterly Journ. Pure and Appl. Math. 31 (1900), 264–314.
- M. F. Vignéras, L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL, Astérisque 61, 235–249 (1979).
- I. Vardi, Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal. 19, 493–507 (1988).
- E.T.Whittaker and G.N.Watson, "A course of modern analysis", CUP.