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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 $

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  • $\begingroup$ What is $k$ here? $\endgroup$ – Daniel Fischer Nov 5 '14 at 18:55
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Based on the following plot of $1/\Gamma(s)$ for $s \in [-4,5]$, I would conjecture that your claim is not correct:

enter image description here

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

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