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Touchard PolynomialsThe Touchard polynomials comprise a polynomial sequence of binomial type defined by -
\left\{\begin{matrix} n \\ k \end{matrix}\right\}x^k 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.) The value at 1 of the nth Touchard polynomial is the nth Bell number, i.e., the number of partitions of a set of size n: -
If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(Xn) = Tn(λ). Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities: -
The Touchard polynomials make up the only polynomial sequence of binomial type in which the coefficient of the 1st-degree term of every polynomial is 1. The Touchard polynomials satisfy the recursion -
In case x = 1, this reduces to the recursion formula for the Bell numbers. The generating function of the Touchard polynomials is -
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