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:

    An educational method that shall have liberty as its basis must intervene to help the child to a conquest of liberty. That is to say, his training must be such as shall help him to diminish as much as possible the social bonds which limit his activity.
    Maria Montessori (1870–1952)

    What comes over a man, is it soul or mind
    That to no limits and bounds he can stay confined?
    You would say his ambition was to extend the reach
    Clear to the Arctic of every living kind.
    Why is his nature forever so hard to teach
    That though there is no fixed line between wrong and right,
    There are roughly zones whose laws must be obeyed?
    Robert Frost (1874–1963)