I was just fooling around on Desmos and entered the function $f_{n}(x)=\frac{n!}{x!(n-x)!}$. When I set $n=0$, i.e. $f_{0}(x)=\frac{1}{x!(-x)!}$, I found that the resulting function looked strikingly similar to the sinc$(\pi x)$ function. When I entered $g(x) = \frac{\sin(\pi x)}{\pi x}$ and graphed the difference function $f(x)-g(x)$, the result was, well, zero, supposedly pointing towards the idea that $$\frac{1}{x!(-x)!} = \frac{\sin(\pi x)}{\pi x}$$ I am aware of the $\Gamma$ function but not very experienced with it. Is there a method by which we can formally arrive at this result, and is there a more general connection between $\Gamma$ and sinc? Any insights would be very much appreciated!
This formula is very similar to the reflection formula, which is $$\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin \pi x}$$ Taking the reciprocal and multiplying by $\pi$, we get $$\frac{\pi}{\Gamma(x)\Gamma(1-x)}=\sin (\pi x)$$