I recently encountered the following, but I am unsure under what conditions for $n$ it is true:
$$\frac{\Gamma(\frac{n}{2}-2)}{\Gamma(\frac{n}{2})}=\frac{1}{(\frac{n}{2}-2)(\frac{n}{2}-1)}$$
I understand how this would be true whenever $n$ is an even integer that's greater than 4 (since I can apply factorial properties), but is it true in general for a real number $n$? If so, is there a general way to determine the value of the ratio of any two gamma functions?