Is it possible to define $$ \frac{\Gamma(a+b-1)}{\Gamma(a-1)} = \frac{\Gamma(1-(a+b))}{\Gamma(1-a)} $$
$$ s.t. a+b>0 \\ a,i\in \mathbb N $$
I know $$\Gamma(x)$$ is defined when $$x>0,x\in \mathbb N $$ But suppose $x$ is a complex number, can the gamma function be defined for negative values?