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I know the Gamma function has a first order pole at the negative integers, and the residue at -k is (-1)^k/k!

So now I want to compute Gamma(-k)/Gamma(-l) , k and l positive integers, and I have a feeling it should be [ (-1)^k/k! ] / [ (-1)^l/l! ] =(-1)^{k+l} l!/k!

What would be an easy way to prove that...?

I thought about considering Gamma(x)/Gamma(x-l+k) and then doing a Laurent expansion of the numerator and denominator around x=-k

This would be

[tex]\frac{\frac{(-1)^k}{(x+k)k!}+reg.}{\frac{(-1)^l}{(x-l+2k)l!}+reg.}[/tex]

However this doesnt seem to make much sense being divergent for x->-k ...

Any tips?

-Pere

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# Gamma of negative integers

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