From the discussion here, it seems that a generalization of the factorial can be written as
$$z!\equiv\Gamma\left(z+1\right)\mid z\in\mathbb{C}-\left\{-1,-2,\dots\right\}$$
We know that in the case where $z$ is real, we can simply write out
$$z! = \prod_{j=1}^z k$$
Is there a way to do the same for $z!$ when $z$ is complex? That is, can it be said, for example that
$$z!=z\left(z-1\right)\left(z-2\right)\cdots$$
My guess is no since the above product won't "terminate" as it does for real $z$. Additionally, from the extended definition using the Gamma function above, why do we exclude only the negative integers and not the entire negative real line?