Let $r_1...r_k$ be the $k$ roots unity or solutions to the expression $x^k = 1$
What is the expression:
$$\frac{1}{x^{k-1}}\frac{1}{\Gamma(-xr_1)}\frac{1}{\Gamma(-xr_2)}\ldots\frac{1}{\Gamma(-xr_k)}$$
equal to?
For the case of $k = 2$:
$$\frac{1}{x}\frac{1}{\Gamma(-x)}\frac{1}{\Gamma(x)}= \frac{-\sin(\pi x)}{\pi}$$