Is there a generalization of Weierstrass infinite product to the function
$$ \frac{1}{g(x,a)}=xa^{-x}e^{\Psi (a) x}\prod_{n=0}^{\infty}\left(1-\frac{x}{c_{n}(a)}\right)e^{\frac{x}{n+a}}$$
$ \Psi(a) $ is the digamma function
Whenever $a=1$ then $ g(x,1)=\Gamma(x) $
The product if over the poles of the function $ g(x,a)$
So, is this a generalization of a weierstrass product of $ g(x,a) $?