  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 $\Gamma_{a,r}(s)$.
