What to do when $x$ in $\Gamma(x)$ is a negative integer?

Question

I have the following likelihood calculation:

\begin{align}\mathcal{L}(s|\alpha) = \sum_{i=1}^O\Biggl\{\ln\frac{ \Gamma(\alpha_0 )}{ \Gamma( B )}- \sum_{k=1}^K \ln \Gamma(\hat{\mathcal{S}}_k^i+1) + \ln \biggl[ \sum_{k=1}^K \Bigl( \ln \Gamma(\hat{\mathcal{S}}_k^i + \alpha_k) - \ln \Gamma(\alpha_k) \Bigr) \biggr] \Biggr\}\end{align}

There are several $\Gamma$ functions in there. $\Gamma$ is not defined for negative integers. When one of the $x$ values in $\Gamma(x)$ is a negative integer, what should I do? Can I add a small number to $x$ (e.g. $x = x+1e^{-131} \quad \text{if}\ x\ \text{is a negative integer}$?

---

Edit: https://math.stackexchange.com/a/263755/96592 could be the answer? So if $x$ is negative $\Gamma(x) = \frac{\Gamma(x+\epsilon)}{\Gamma(\epsilon)}$ with $\epsilon$ close to $0$? Not sure if I understand that correctly.

Answer 1

$\Gamma(x)$ has poles at non-positive integers, so you won't be able to solve the problem by adding a small value to $x$, since the smaller the value you use, the larger (or more negative depending on if x is odd or even) $\Gamma(x)$ will be.

In other words, for even $x \in \mathbb{Z}_{\leq 0}$

$\lim_{\epsilon \rightarrow 0^+}\Gamma(x+\epsilon)=\infty$ and $\lim_{\epsilon \rightarrow 0^-}\Gamma(x+\epsilon)=-\infty$

and vice versa for odd $x \in \mathbb{Z}_{\leq 0}$

Answer 2

1. The gamma function $\Gamma(x)$ is positive on the intervals $(0,+\infty)$ and $(-2k,1-2k)$ for $k\in\mathbb{N}=\{1,2,\dotsc\}$. See Figure 3.1 on page 44 in the book [4] below.
2. The gamma function $\Gamma(x)$ is negative on the intervals $(1-2k,2-2k)$ for $k\in\mathbb{N}$. See Figure 3.1 on page 44 in the book [4] below.
3. The gamma function $\Gamma(z)$ is single-valued and analytic over the punctured complex plane $\mathbb{C}\setminus\{1-k,k\in\mathbb{N}\}$. See Item 6.1.3 on page 255 in the handbook [1] below.
4. The gamma function $\Gamma(z)$ has simple poles in the left half-plane at the points $1-k$ and the residue at $1-k$ is $\frac{(-1)^{k-1}}{(k-1)!}$ for $k\in\mathbb{N}$. See page 44 in the book [4] below.
5. The reciprocal $\frac{1}{\Gamma(z)}$ is an entire function possessing simple zeros at the points $1-k$ for $k\in\mathbb{N}$. See Item 6.1.3 on page 255 in the handbook [1] below.
6. The limit \begin{equation*} \lim_{z\to-n}[(z+n)\Gamma(z)]=\frac{(-1)^n}{n!}, \quad n\in\mathbb{N}_0=\{0,1,2,\dotsc\} \end{equation*} is useful. See page 44 in the book [4] below.
7. 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\mathbb{N}_0, \quad n,q\in\mathbb{N} \end{equation*} is useful. See the papers [2, 3, 5] below.

References

1. M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992.
2. 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.
3. 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.
4. N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1996; available online at http://dx.doi.org/10.1002/9781118032572.
5. 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.