In Complex Analysis by Ahlfors (p198), the author starts with the functional $$ G(z) = \prod_1^\infty \left( 1 + \frac{z}{n} \right) e^{-z/n} $$ and goes on to state that we may obviously write $$ G(z-1)=ze^{\gamma(z)}G(z) $$ where $\gamma(z)$ is entire. I fail to see how he comes up with this construction, other than that it provides a zero at $z=0$. How did he come up with this form?
He goes on to show that $\gamma$ is constant and equal to Euler's constant, 0.57722.
$G$ is an entire function with zeros exactly at $z=-1, -2, -3, \dotsc$, and all zeros are simple. It follows that (as already mentioned in a comment) $$ \frac{G(z-1)}{zG(z)} $$ is an entire function without zeros, and therefore equal to $e^{\gamma(z)}$ for some entire function $\gamma$.