## A simple derivation of Stirling’s asymptotic series.(English)Zbl 0615.05010

In this note a relatively simple derivation is given for Stirling’s asymptotic expansion $n!=e^{-n} n^ n \sqrt{2\pi n}\exp \{\frac{1}{12n}-\frac{1}{360n^ 3}+\frac{1}{1260n^ 5}+...\}.$ The proof starts with a derivation of the duplication formula $$\Gamma (2n)=(1/\sqrt{\pi})\Gamma (n)\Gamma (n+)2^{2n-1}$$ invoking $$\Gamma ()=\sqrt{\pi}$$ along the way. Then F(n) is defined both by F(n)$$\sqrt{2\pi n}n^{n-1}e^{-n}=\Gamma (n)$$ and by $$F(n)=\exp \sum^{\infty}_{k=1}a_ k/n^ k$$. A recursion is then found for the $$a_ k$$ from the duplication formula. It is also shown that $$k(k+1)a_ k=B_{k+1}$$, the $$(k+1)st$$ Bernoulli number, and a family of recursions for the Bernoulli numbers is indicated.

### MSC:

 05A15 Exact enumeration problems, generating functions 33B15 Gamma, beta and polygamma functions

### Keywords:

duplication formula; Bernoulli number; recursions
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