How can I use the Stirling's approximation to approximate factorials?

Question

I'd like to exploit Stirling's approximation during the symbolic manipulation of an expression. Essentially, I want replace Factorial[n] with n^n E^-n Sqrt[2 \Pi n] everywhere in a large expression.

How is it possible to achieve that?

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As an example, I'd like to approximate a sum like this

$$ \frac{n!}{(n+m_1)! (n-m_1)!} + \frac{n!}{(n+m_2)! (n-m_2)!} + \frac{n!}{ (n+m_3)! (n-m_3)!} $$ to $$ \frac{F(n)}{F(n + m_1)Fun(n - m_1)} + \frac{F(n)}{F(n + m_2) F(n - m_2)} + \frac{F(n)}{F(n + m_3) F(n - m_3)}, $$ where $$ F(n) := \sqrt{2 \pi n} \, e^{-n} n^n $$ is the Stirling approximation of $n!$.

Answer 1

Note: updated code as per comments below. I believe this works:

Subscript[m, 1] = 2;
Subscript[m, 2] = 3;
Subscript[m, 3] = 4;
myf1[n_] = 
  n!/((n - Subscript[m, 1])! (n + Subscript[m, 1])!) + 
   n!/((n - Subscript[m, 2])! (n + Subscript[m, 2])!) + 
   n!/((n - Subscript[m, 3])! (n + Subscript[m, 3])!);
myf2[n_] = (myf1[n] /. (n_)! -> (Sqrt[2 Pi n] Exp[-n] n^n))
myf2[n] // TeXForm
myf1[100] // N
myf2[100] // N

$$ \frac {e^n (n - 4)^{\frac {7} {2} - n} n^{n + \frac {1} {2}} (n + 4)^{-n - \frac {9} {2}}} {\sqrt {2 \pi }} + \frac {e^ n (n - 3)^{\frac {5} {2} - n} n^{n + \frac {1} {2}} (n + 3)^{-n - \frac {7} {2}}} {\sqrt {2 \pi }} + \frac {e^n (n - 2)^{\frac {3} {2} - n} n^{n + \frac {1} {2}} (n + 2)^{-n - \frac {5} {2}}} {\sqrt {2 \pi }} $$

2.923181964355016`*10^-158
2.925623558480809`*10^-158

Answer 2

As mentioned in the comments, just use Series:

series = Series[
    n!/(n+m1)!/(n-m1)!+n!/(n+m2)!/(n-m2)!+n!/(n+m3)!/(n-m3)!,
    {n, Infinity, 2}
];
series //TeXForm

$\exp \left((1-\log (n)) n+O\left(\left(\frac{1}{n}\right)^5\right)\right) \left(\frac{3 \sqrt{\frac{1}{n}}}{\sqrt{2 \pi }}+\frac{\left(-4 \sqrt{2} \operatorname{m1}^2-4 \sqrt{2} \operatorname{m2}^2-4 \sqrt{2} \operatorname{m3}^2-\sqrt{2}\right) \left(\frac{1}{n}\right)^{3/2}}{8 \sqrt{\pi }}+O\left(\left(\frac{1}{n}\right)^{5/2}\right)\right)$

If you want a normal expression, just use Normal:

Normal @ series //TeXForm

$e^{n (1-\log (n))} \left(\frac{-4 \sqrt{2} \operatorname{m1}^2-4 \sqrt{2} \operatorname{m2}^2-4 \sqrt{2} \operatorname{m3}^2-\sqrt{2}}{8 \sqrt{\pi } n^{3/2}}+\frac{3}{\sqrt{2 \pi } \sqrt{n}}\right)$

Answer 3

Another way of doing what you ask, illustrating the use of Block. Define the expression in which to do the replacements

expr = Sum[n!/((n + m[i])! (n - m[i])!), {i, 1, 3}]
(* n!/((n - m[1])! (n + m[1])!) + n!/((n - m[2])! (n + m[2])!) + 
 n!/((n - m[3])! (n + m[3])!) *)

Do the replacement

Block[{Factorial = Sqrt[2 π #] Exp[-#] #^# &}, expr]
(* (E^n n^(
  1/2 + n) (n - m[1])^(-(1/2) - n + m[1]) (n + m[1])^(-(1/2) - n - 
   m[1]))/Sqrt[2 π] + (
 E^n n^(1/2 + n) (n - m[2])^(-(1/2) - n + m[2]) (n + m[2])^(-(1/2) - 
   n - m[2]))/Sqrt[2 π] + (
 E^n n^(1/2 + n) (n - m[3])^(-(1/2) - n + m[3]) (n + m[3])^(-(1/2) - 
   n - m[3]))/Sqrt[2 π] *)