Weierstrass product representations of multiple gamma and sine functions

Abstract

It is well known that the Weierstrass product representation of the Barnes multiple gamma function Γ_r(z) can be calculated concretely. However, there has been no study on its explicit formulation. In this paper, its simple formulation is achieved. It is applicable to the Weierstrass product representation of the Vignéras multiple gamma function also. Moreover, the Weierstrass product representation of the Kurokawa multiple sine function S_r(z) is also formulated explicitly.