Barnes G-function

In mathematics, the Barnes G-function G ( z ) {\displaystyle 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. It can be written in terms of the double gamma function. Formally, the Barnes G-function is defined in the following Weierstrass product form: where γ {\displaystyle \,\gamma } is the Euler–Mascheroni constant, exp(x) = e is the exponential function, and Π {\displaystyle \Pi } denotes multiplication (capital pi notation). The integral representation, which may be deduced from the relation to the double gamma function, is As an entire function, G {\displaystyle G} is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below. The Barnes G-function satisfies the functional equation with normalization G ( 1 ) = 1 {\displaystyle G(1)=1} . Note the similarity between the functional equation of the Barnes G-function and that of the Euler gamma function: The functional equation implies that G {\displaystyle G} takes the following values at integer arguments: In particular, G ( 0 ) = 0 , G ( 1 ) = 1 {\displaystyle G(0)=0,G(1)=1} and G ( n ) = s f ( n − 2 ) {\displaystyle G(n)=sf(n-2)} for n ≥ 1 {\displaystyle n\geq 1} , where s f {\displaystyle sf} is the superfactorial. and thus where Γ ( x ) {\displaystyle \,\Gamma (x)} denotes the gamma function and K {\displaystyle K} denotes the K-function. In general, K ( z ) G ( z ) = e ( z − 1 ) ln ⁡ Γ ( z ) {\displaystyle K(z)G(z)=e^{(z-1)\ln \Gamma (z)}} for all complex z {\displaystyle z} . The functional equation G ( z + 1 ) = Γ ( z ) G ( z ) {\displaystyle G(z+1)=\Gamma (z)\,G(z)} uniquely defines the Barnes G-function if the convexity condition, is added.