Lower bound on a ratio of factorials

Question

I'm trying to get a lower bound on a ratio of factorials like $$ \frac{(x-Ay+C)!(x-y)!}{(x-By+D)!x!} $$ where $$ x,y,A,B,C,D \in \mathbb{N}, \quad By-D < x, \quad A < B, \quad C < D $$

I can also make assumptions on the range these variables fall in, but I'm not going to list them, because I'd rather not have anyone do the work. I'm really just looking for references or tips on a variety of ways to do this.

Edit: So far, we've seen Stirling's approximation and Ramanujan's approximation. I'm still interested in any other references for approximating or lower bounding factorials in general.

Answer 1

One can use the Stirling bounds

$$\sqrt{2\pi n}\left(\frac ne\right)^n<n!<\sqrt{e^2n}\left(\frac ne\right)^n,\quad n>1$$

Which gives the lower bound

$$\frac{(x-Ay+C)!(x-y)!}{(x-By+D)!x!}>\frac{2\pi}{e^2}\frac{(x-Ay+C)^{x-Ay+C+\frac12}}{(x-By+D)^{x-By+D+\frac12}}\frac{(x-y)^{x-y+\frac12}}{x^{x+\frac12}}e^{(A+1-B)y+D-C}$$