Gamma function of negative argument

Question

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?

Answer 1

Your relation /would/ hold if $\Gamma$ were continuous at $-1$. It is not, however: intuitively, we cannot take the factorial of negative integers.

Answer 2

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.

Answer 3

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

1. 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.
2. 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.
3. 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.

Answer 4

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