Barnes G

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The Barnes $G$ function is defined by the following Weierstrass factorization: $$G(1+z)=(2\pi)^{\frac{z}{2}} \exp \left( - \dfrac{z+z^2(1+\gamma)}{2} \right) \displaystyle\prod_{k=1}^{\infty} \left\{ \left( 1+\dfrac{z}{k} \right)^k \exp \left( \dfrac{z^2}{2k}-z \right) \right\},$$ where $\exp$ denotes the exponential function and $\gamma$ denotes the Euler-Mascheroni constant.


Barnes G at z+1 in terms of Barnes G and gamma
Barnes G at positive integer


The theory of the $G$-function by E.W. Barnes