derivatives of the inverse gamma function at negative integers??

Question

what are the derivatives of the function $ \frac{1}{\Gamma(s)} $ at 0 and negative integers ?? i believe that $$ \frac{ d^{k}}{ds^{k}}\frac{1}{\Gamma(s)}=0 $$ for

$ s=0,-1,-2,-3,-4 $

Answer 1

Based on the following plot of $1/\Gamma(s)$ for $s \in [-4,5]$, I would conjecture that your claim is not correct:

With more computation from Mathematica, we have for the first derivative $$\frac{d}{ds} \left[ \frac{1}{\Gamma(s)} \right] = (-1)^s s!, \quad s = 0, -1, -2, \ldots.$$