A class of generalized gamma functions

Abstract

In this paper, we study the holomorphic function defined by the infinite product ${\Gamma }_{a,r}(s)={\prod }_{n\ge 0}(1+\frac{1}{a+nr}{)}^s(1+\frac{s}{a+nr}{)}^{-1}$ which generalize Euler's definition in the sense that $\Gamma (s)=\frac{{\Gamma }_{1,1}(s)}{s}$. We obtain analogues of the Gauss multiplication formula, complement formula for functions ${\Gamma }_{a,r}(s)$.

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