Touchard polynomials

From Academic Kids

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

<math>p_n(x)=\sum_{k=1}^n S(n,k)x^k=\sum_{k=1}^n

\left\{\begin{matrix} n \\ k \end{matrix}\right\}x^k<math>

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:

<math>T_n(1)=B_n.<math>

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:

<math>T_n(\lambda+\mu)=\sum_{k=0}^n {n \choose k} T_k(\lambda) T_{n-k}(\mu).<math>

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

<math>T_{n+1}(x)=x\sum_{k=0}^n{n \choose k}T_k(x).<math>

In case x = 1, this reduces to the recursion formula for the Bell numbers.

The generating function of the Touchard polynomials is

<math>\sum_{n=0}^\infty {T_n(x) \over n!} t^n=e^{x\left(e^t-1\right)}.<math>
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