Is there a meaningful way to assign values $\Gamma^\star(z)$ related to the values of the gamma function $\Gamma(z) = \int_0^\infty {e^{-t}t^{-z}dt}$ such that $\prod\limits_{ \Bbb Z^-} \Gamma^\star (z) = \Gamma^\star(1)$?
Intuitively, I feel like they should because of the recursive relation and fact that $\Gamma(0) = 1$. Perhaps $\Gamma^\star$ is related to residues or Cauchy principle values?
Otherwise, is there a good proof that $\lim\limits_{\epsilon \to 0}({\prod\limits_{\Bbb Z^-}{\Gamma(z + \epsilon)}}) = 1$ that doesn't just use the recursive definition?
I wasn't sure of the right terminology to use so please let me know if this is too soft and I won't mind closing it.