I am trying to find a clever way to compute the quotient of two gamma functions whose inputs differ by some integer. In other words, for some real value $x$ and an integer $n < x$, I want to find a way to compute
$$ \frac{\Gamma(x)}{\Gamma(x-n)} $$
For $n=1$, the quotient it is simply $(x-1)$ since by definition
$$ \Gamma(x) = (x – 1)\Gamma(x-1) $$
For $n=2$, it is also simple:
$$ \frac{\Gamma(x)}{\Gamma(x-2)} =
\frac{(x-1)\Gamma(x-1)}{\Gamma(x-2)} =
(x-1)(x-2)$$
If we continue this pattern out to some arbitrary $n$, we get
$$ \frac{\Gamma(x)}{\Gamma(x-n)} = \prod_i^n (x-i)$$
Obviously I am floundering a bit here. Can anyone help me find an efficient way to compute this quotient besides directly computing the two gamma functions and dividing?
I am also okay if an efficient computation can be found in log space. Currently I am using a simple approximation of the log gamma function and taking the difference. This was necessary because the gamma function gets too big to store in any primitive data type for even smallish values of $x$.
Best Answer
You can write the expression in terms of the Pochhammer symbol as
$$ (x-1)_n = \frac{\Gamma(x)}{\Gamma(x-n)}. $$