Approximation of the factorial function

Question

I'm using the term factorial function as $\gamma(x+1)$ on the sense that I'm taking all real number in count.

I have seen many approximations of the factorial function for positive values, for example the Stirling approximation which starts approaching the function after 1 and the Ramanujan approximation which is very close for all positive $x$.

But I've never seen an approximation that works with negative values. I was wondering if anyone know one, it would be even better if it was on a closed form.

Any thoughts would be really appreciated!!

Answer 1

Gamma function discontinuous for all non-positive integers, it's natural to have no direct approximations for $x \le 0$.

For $\lfloor x \rfloor =-n$ and $\{ x \}=x+n \in (0,1)$,

\begin{align*} \Gamma(x) &= \frac{\Gamma(x+1)}{x} \\ &= \frac{\Gamma(x+n)}{x(x+1)(x+2)\ldots (x+n-1)} \\ &= \frac{\Gamma(\{ x \})}{(x)_{n}} \end{align*}