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

Namias, Victor

Am. Math. Mon. 93, 25-29 (1986).

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