How to compute $\frac{\Gamma(n-\alpha)}{\Gamma(\alpha)}?$

Question

How to simplify the gamma function: $$\frac{\Gamma(n-\alpha)}{\Gamma(\alpha)}?$$ where $n\in \mathbb{Z}$ and $\alpha$ some real number.

Answer 1

Hint: If you assume $0<\alpha<1$, you should be able to express this in terms of $\Gamma(\alpha/2)$ and $\Gamma(1-\alpha/2)$ using the fact that $n!= n(n-1)!$. Once you have that, you may be able to use the reflection formula $$\Gamma(z)\Gamma(1-z) =\frac {\pi}{\sin \pi z}$$This is only speculation on my part, but I think it is a path worth exploring.

Addendum: The reflection formula means you could write $$\frac{1}{\Gamma(1-z)}= \frac{1}{\pi}\Gamma(z)\sin \pi z$$ so $$\frac{\Gamma(z)}{\Gamma(1-z)}= \frac {1}{\pi}\Gamma(z)^2\sin \pi z$$