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A new Stirling series as continued fraction. (English) Zbl 1213.33007
Summary: We have the well-known Stirling’s formula $n!\sim \sqrt{2\pi n}(\frac ne)^n$. This formula is the first approximation of the stirling series: $$n~\sim\sqrt{2\pi n} \bigg(\frac ne\bigg)^n \exp (\bigg( \frac{1}{12n}- \frac{1}{360n^3}+ \frac{1}{1260^{n^5}}- \cdots\bigg).$$ The author has formulated the following new Stirling series as a continued fraction $$ n! \sim \sqrt{2\pi n}\bigg( \frac{n}{e}\bigg)^n\exp \cfrac1\\ 12n+\cfrac \frac25\\ n+ \cfrac \frac{53}{210}\\ n+\cfrac \frac{195}{371}\\ n+\cfrac \frac{22999}{22737}\\ n+\ddots \endcfrac $$

33B15Gamma, beta and polygamma functions
11B73Bell and Stirling numbers
40A15Convergence and divergence of continued fractions
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
Full Text: DOI
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