Limits defined for negative factorials (i.e. $(-n)!,\space n\in\mathbb{N}$)

Question

I apoligize if this is a stupid/obvious question, but last night I was wondering how we can compute limits for factorials of negative integers, for instance, how do we evaluate:

$$\lim_{x\to-3}\frac{x!}{(2x)!}=-120$$

Neither $x!$, nor $(2x)!$ are defined for $x\in\mathbb{Z}^{-}$, and indeed, both are singularities according to the graph of $\Gamma(x+1)$.

The book I am reading calculates this using a previously shown identity that:

$$F\left(\left.{1-c-2n,-2n \atop c}\right|-1\right)=(-1)^{n}\frac{(2n)!}{n!}\frac{(c-1)!}{(c+n-1)!},\space\forall n\in\mathbb{Z}^{*}$$

And then, the more general Kummer's Formula:

$$F\left(\left.{a,b \atop 1+b-a}\right|-1\right)=\frac{(b/2)!}{b!}(b-a)^{\underline{b/2}}$$

It then shows that they would only produce consistent results if:

$$(-1)^{n}\frac{(2n)!}{n!}=\lim_{b\to-2n}{\frac{(b/2)!}{b!}}=\lim_{x\to-n}{\frac{x!}{(2x)!}},\space n\in\mathbb{Z}^{*}$$

It then gives the example of $n=3$, proving that:

$$\lim_{x\to-3}{\frac{x!}{(2x)!}}=-\frac{6!}{3!}=-120$$

However, using Wolfram|Alpha, I can see that there are other such limits defined (such as $\lim_{x\to-3}{\frac{x!}{(8x)!}}=-103408066955539906560000$.

Without using the hypergeometric series, how could we evaluate limits such as these?

Again, sorry if this is a stupid question, thanks in advance!

Answer 1

You want to compute $\displaystyle \lim_{x\to -n} \frac {\Pi(x)}{\Pi(mx)}$ when $x$ is near a negative integer.
$\Pi$ is the 'natural' extension of the factorial : $\Pi(n)=n!$ and $\Pi(z)=\Gamma(z+1)$ (see Wikipedia)

In this form the "Euler's reflection formula" becomes simply (for $\operatorname{sinc}(z)=\frac{\sin(\pi z)}{\pi z}$) : $$\Pi(-z)\Pi(z)=\frac 1{\operatorname{sinc}(z)}$$

$$ \lim_{x\to -n}\ \frac {\Pi(x)}{\Pi(mx)}=\lim_{x\to -n}\frac {\Pi(-mx)\operatorname{sinc}(-mx)}{\Pi(-x)\operatorname{sinc}(-x)}$$ $$ =\lim_{t\to n}\frac {\Pi(mt)\operatorname{sinc}(mt)}{\Pi(t)\operatorname{sinc}(t)}$$

It remains to prove that $\ \lim_{t\to n} \frac {\operatorname{sinc(mt)}}{\operatorname{sinc(t)}}=\frac {(-1)^{(m-1)n}}m$ (you may use l'Hôpital's rule for that) and to conclude!

Answer 2

Using anon's "pole" idea, with definition $x! := \Gamma(x+1)$ we have: $$\begin{align} x! &= \Gamma(x+1) = \frac{1}{2}\;\frac{1}{x+3}+O(1)\qquad\text{as $x \to -3$}, \\ (2x)! &= \Gamma(2x+1) = -\frac{1}{240}\;\frac{1}{x+3} + O(1)\qquad\text{as $x \to -3$}, \\ \frac{x!}{(2x)!} &= \frac{1/2}{-1/240}+O(x+3)\qquad\text{as $x \to -3$}, \\ \frac{x!}{(2x)!} &\to -120\qquad\text{as $x \to -3$}. \end{align}$$

Answer 3

Let me add to the above, that while $\Gamma(-n)$ for a positive integer $n$ is undefined, let $m$ be such an integer as well, and then the ratio $\Gamma(-n)/\Gamma(-m)$ is well defined, and the Euler reflection formula above leads to its value being equal to $\Gamma(m+1)/\Gamma(n+1)(-1)^{n-m}$. This shows, by the way that the ratio mentioned at the beginning of the this sequence, effectively $(-3)!/(-6)!$ is $-60$, and not as suggested.

Answer 4

I think that the limit formulas \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*} and \begin{equation}\label{polygamma-limit-eq} \lim_{z\to-k}\frac{\psi(nz)}{\psi(qz)}=\frac{q}{n}, \quad k\in\{0,1,2,\dotsc\} \quad n,q\in\{1,2,\dotsc\} \end{equation} give a perfect answer. One can find alternative proofs of these limit formulas 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.