I have a product of the following form:
$$ P = (N-\alpha)(N-2\alpha)\cdots(N-k\alpha) $$
where $k$ is an integer between $1$ and $N-1$, and $\alpha$ is a real number in $(0,1]$.
Clearly, for $\alpha=1$,
$$ P = \frac{(N-1)!}{(N-k-1)!} $$
Now, in general I can write:
$$ P = \alpha^k \left(\frac{N}{\alpha}-1\right) \left(\frac{N}{\alpha}-2\right) \cdots \left(\frac{N}{\alpha}-k\right) $$
This looks like a factorial, but $\frac{N}{\alpha}$ is not an integer in general. I tried rounding it up and using the factorial function anyway. This gave results in the right ballpark (at least for the values I tried), but they were still off by quite a bit, which I assume is because the rounding error is being multiplied.
I am not very familiar with the Gamma function, but looking at its definition and relation to the factorial function, I took a guess and wrote:
$$ P = \alpha^k \frac{\Gamma\left(\frac{N}{\alpha}\right)}{\Gamma\left(\frac{N}{\alpha}-k\right)} $$
This gave exactly the same results as the factorial function with rounding.
Is this a proper way of using the Gamma function?
Is there a more accurate way to write $P$ using it or anything else?
You can write $P$ with the Gamma function, but more simply using falling powers. There are several notations, but using the one from the linked wiki article, $(n)_k=n(n-1)(n-2)\cdots (n-k+1)$. Hence, $$P=\alpha^k \left(\frac{N}{\alpha}-1\right)_k$$