- Barnes G-function
In
mathematics , the Barnes G-function (typically denoted "G"("z")) is a function that is an extension ofsuperfactorial s to thecomplex number s. It is related to theGamma function , theK-function and theGlaisher-Kinkelin constant , and was named aftermathematician Ernest William Barnes . [ E.W.Barnes, "The theory of the G-function", "Quarterly Journ. Pure and Appl. Math." 31 (1900), 264-314.]Formally, the Barnes G-function is defined (in the form of a
Weierstrass product ) as:
where γ is the
Euler-Mascheroni constant .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::
and thus
:
where Γ denotes the
Gamma function and "K" denotes theK-function . The difference equation uniquely defines the G function if the convexity condition: is added [M. F. Vignéras, "L'équation fonctionelle de la fonction zêta de Selberg du groupe mudulaire SL", Astérisque 61, 235-249 (1979).] .The
difference equation for the G function and thefunctional equation for theGamma function yield the followingfunctional equation for the G function, originally proved byHermann Kinkelin ::
Multiplication formula
Like the Gamma function, the G-function also has a multiplication formula [I. Vardi, "Determinants of Laplacians and multiple gamma functions", SIAM J. Math. Anal. 19, 493-507 (1988).] ::where is a constant given by:
:
Here is the derivative of the
Riemann zeta function and is theGlaisher-Kinkelin constant .Asymptotic Expansion
The function has the following asymptotic expansion established by Barnes::
Here the are the
Bernoulli numbers and is theGlaisher-Kinkelin constant . (Note that somewhat confusingly at the time of Barnes [ E.T.Whittaker and G.N.Watson, "A course of modern analysis", CUP.] theBernoulli number would have been written as , but this convention is no longer current.)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.References
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