Gamma function of negative argument Is there any relation between the limiting behaviour of $\Gamma({\epsilon})$ and $\Gamma(-1+{\epsilon})$?
I have seen the relation such as $\Gamma(-1+{\epsilon})$ $=$ $\Gamma({\epsilon})/(-1+{\epsilon})$. I think it is basically wrong? But does there exist such a similar relation?
 A: The relation $\Gamma(-1+\epsilon) = \Gamma({\epsilon})/(-1+{\epsilon})$ is true so long as $\epsilon$ is not a negative integer (so that $-1+\epsilon$ will then also not be a negative integer) since the gamma function is extended to the complex plane minus the negative integers by using the relation $\Gamma(z)=\Gamma(z+1)/z$ or by using analytic continuation.
Thus, you can say something about the limiting behaviour of $\Gamma(\epsilon)$ and $\Gamma(-1+\epsilon)$, in that you can say that 
$$\lim_{\epsilon\to 0} \frac{\Gamma(-1+\epsilon)}{\Gamma(\epsilon)} = \lim_{\epsilon\to 0} \frac{1}{-1+\epsilon} = -1.$$
Note that the fact that $\Gamma(z)$ is not defined at $-1$ does not affect this, since for the limit, we are only interested in the values of the function close to $-1$.
In other words, $|\Gamma(z)\vert$ tends to infinity "at the same rate" as $z\to 0$ or as $z\to -1$, and similar results could be proved at any negative integer.
A: I think that the limit formula
\begin{equation*}%\label{gamma-limit-eq}
\lim_{z\to-k}\frac{\Gamma(nz)}{\Gamma(qz)}=(-1)^{(n-q)k}\frac{q}{n}\frac{(qk)!}{(nk)!}, \quad k\in\{0,1,2,\dotsc\} \quad n,q\in\{1,2,\dotsc\}
\end{equation*}
is a perfect answer. One can find alternative proofs of this limit formula in the papers [1, 2, 3] below.
References

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*A. Prabhu and H. M. Srivastava, Some limit formulas for the Gamma and Psi (or Digamma) functions at their singularities, Integral Transforms Spec. Funct. 22 (2011), no. 8, 587--592; available online at https://doi.org/10.1080/10652469.2010.535970.

*F. Qi, Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat 27 (2013), no. 4, 601--604; available online at http://dx.doi.org/10.2298/FIL1304601Q.

*L. Yin and L.-G. Huang, Limit formulas related to the $p$-gamma and $p$-polygamma functions at their singularities, Filomat 29 (2015), no. 7, 1501--1505; available online at https://doi.org/10.2298/FIL1507501Y.

A: Your relation /would/ hold if $\Gamma$ were continuous at $-1$. It is not, however: intuitively, we cannot take the factorial of negative integers.
A: Regarding numerical computation of Gamma Function for negative real numbers, $\Gamma(x)$, can one use the relation:
$$\frac{1}{\Gamma(x)}\!=\!\sum\limits^{+\infty}_{m\!=\!1}c_{m}x^{m}$$
where the only stated restriction is $\vert x\vert \leq \infty$? This formula appears in Abramowitz and Stegun (Handbook for Mathematical Functions and Tables, page 256, formula # 6.1.34
