# Lower bound on a ratio of factorials

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.

• Stirling approximations everywhere? – Simply Beautiful Art Sep 22 '17 at 0:39
• I think it's just a ton of plugging in of numbers. Stirling approximations work. Ramanujan's factorial approximation also works as well. – Stone Sep 22 '17 at 0:45
• Thank you both! I should have mentioned, I'm aware of Stirling's and Ramanujan's approximations. I suppose I'm hoping for something like them, but with a multivariable flavor. Unless there's an easy way to extend them to the case of two variables, like above? I might just end up replacing $y$ with some fraction of $x$ (my use case allows me to do so), and use these single variable approximation methods though. But I do need a lower bound... If I recall, these methods give error bounds, so I suppose I would just subtract the appropriate error bound to yield a lower bound? – TheIntern Sep 22 '17 at 0:50
• Just did some reading, it seems like the above is the way to go. It also looks like there's some really great optimal bounds recently proven too. I'm still interested in other approaches though--mostly just for fun. Perhaps a reference on approximating factorials in general? – TheIntern Sep 22 '17 at 0:56

## 1 Answer

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

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

• Ah, of course you can just plug in to get a multivariable result. Thanks! – TheIntern Sep 22 '17 at 1:04
• No problem. @TheIntern – Simply Beautiful Art Sep 22 '17 at 1:05