Bell Number - Properties of Bell Numbers

Properties of Bell Numbers

The Bell numbers satisfy this recursion formula:

They also satisfy "Dobinski's formula":

= the nth moment of a Poisson distribution with expected value 1.

And they satisfy "Touchard's congruence": If p is any prime number then

or, generalizing

Each Bell number is a sum of Stirling numbers of the second kind

The Stirling number is the number of ways to partition a set of cardinality n into exactly k nonempty subsets.

More generally, the Bell numbers satisfy the following recurrence:

The nth Bell number is also the sum of the coefficients in the polynomial that expresses the nth moment of any probability distribution as a function of the first n cumulants; this way of enumerating partitions is not as coarse as that given by the Stirling numbers.

The recurrence relation at the top of this section can be used to show the exponential generating function of the Bell numbers is satisfies the differential equation, from which one can derive

An application of Cauchy's integral formula yields the complex integral representation

Some asymptotic representations can then be derived by a standard application of the method of steepest descent.