What is $\Gamma(-k+\epsilon)$ where $k=0,1,2,...$?

Question

I'm trying to find a general solution to this, using $\Gamma(z+1)=z\Gamma(z)$. My suspicion is that is will be a series involving $\frac{(-1)^{k}}{k!}$, the residues at k, but I could really use a push in the right direction.

Edit: I need it of the form $\frac{(-1)^{k}\epsilon}{k!}+a_{k}+O(\epsilon)$.

Answer 1

The reflection formula for the Gamma function gives you $$ \Gamma \left( -k+\epsilon \right) ={\frac {\pi}{\Gamma \left( k+1- \epsilon \right) \sin \left( \pi\, \left( -k+\epsilon \right) \right) }} = \frac{(-1)^k \pi}{\Gamma(k+1-\epsilon) \sin(\pi \epsilon)} $$