Bell Number - Asymptotic Limit and Bounds

Asymptotic Limit and Bounds

Several asymptotic formulae for the Bell numbers are known. One such is

Here

where W is the Lambert W function. (Lovász, 1993)

Moser and Wyman established the expansion

uniformly for as, where and each and are known expressions in .

In (Berend, D. and Tassa, T., 2010), the following bounds were established:

moreover, if then for all ,

where and $~d(x):= \ln \ln (x+1) - \ln \ln x + \frac{1+e^{-1}}{\ln x}\,.$

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