Main Page | See live article | Alphabetical index

Touchard polynomials

The Touchard polynomials comprise a polynomial sequence of binomial type defined by

where S(n, k) is a Stirling number of the second kind, i.e., it is the number of partitions of a set of size n into k disjoint non-empty subsets. (The second notation above, with { braces }, was introduced by Donald Knuth.)

If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(Xn) = Tn(λ).

This article is a stub. You can help Wikipedia by fixing it.