Touchard Polynomials

The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials, comprise a polynomial sequence of binomial type defined by

$T_0(x) = 1,\qquad T_n(x)=\sum_{k=1}^n S(n,k)x^k=\sum_{k=1}^n \left\{\begin{matrix} n \\ k \end{matrix}\right\}x^k, \quad n > 0,$

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) = T_n(λ), leading to the definition:

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 Rodrigues-like formula:

The Touchard polynomials satisfy the recursion

And

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

Using the Umbral notation Tn(x)=T_n(x),these formulas become:

The generating function of the Touchard polynomials is

And a contour-integral representation is

The Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order: