Residue of f$f(z)=\Gamma \left(\frac{z+1}{a} \right)$

Question

How to find a residue of \begin{align} f(z)=\Gamma \left(\frac{z+1}{a} \right) \end{align} for $a>0$.

I know that the Gamma function has poles for non-positive integers so the polls happen at \begin{align} z_n= -ka-1, \ k=0,1,2,... \end{align}

but no sure how to compute the residue.

Thanks

Answer 1

Hint: $$x\Gamma(x)=\Gamma(x+1)$$

and more generally,

$$(x+a)\dots(x+2)(x+1)(x)\Gamma(x)=\Gamma(x+a+1)$$

Thus, to calculate the residue:

$$(x+a)\Gamma(x)=\frac{\Gamma(x+a+1)}{(x+a-1)\dots(x+2)(x+1)(x)}\to(-1)^a/a!$$

As $x\to-a$.