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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?

Ricky
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No Yeah
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    The Gamma function can be extended to a meromorphic function on the entire complex plane, with simple poles at the non-positive integers $0,-1,-,2,\dots$ having residue $\text{Res}(\Gamma,-n)=\frac{(-1)^n}{n!}$ for all integer $n\geq 0$. So, yes the Gamma function can be defined for (most) negative values, and in fact most complex values. – peek-a-boo Sep 06 '23 at 01:38

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