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}\,.

Read more about this topic:  Bell Number

Famous quotes containing the words limit and/or bounds:

    “There is a limit to the application of democratic methods. You can inquire of all the passengers as to what type of car they like to ride in, but it is impossible to question them as to whether to apply the brakes when the train is at full speed and accident threatens.”
    —Leon Trotsky (1879–1940)

    “Great Wits are sure to Madness near alli’d
    And thin Partitions do their Bounds divide;
    Else, why should he, with Wealth and Honour blest,
    Refuse his Age the needful hours of Rest?”
    —John Dryden (1631–1700)