A class of generalized gamma functions

In this paper, we study the holomorphic function defined by the infinite product $\Gamma_{a,r}(s) =\prod_{n \geq 0} (1 + \frac{1}{a+ nr})^s (1 + \frac{s}{a+nr})^{-1}$ 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 $\Gamma_{a,r}(s)$.