Touchard polynomials

The Touchard polynomials, named after Jacques Touchard, also called the exponential 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.) The value at 1 of the "n"th Touchard polynomial is the "n"th Bell number, i.e., the number of partitions of a set of size "n":

:$T\_n(1)=B\_n.$

If "X" is a random variable with a Poisson distribution with expected value λ, then its "n"th moment is E("X"^"n") = "T"_"n"(λ). Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities:

:$T\_n(lambda+mu)=sum\_\{k=0\}^n\; \{n\; choose\; k\}\; T\_k(lambda)\; T\_\{n-k\}(mu).$

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

:$T\_\{n+1\}(x)=xsum\_\{k=0\}^n\{n\; choose\; k\}T\_k(x).$

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

The generating function of the Touchard polynomials is

:$sum\_\{n=0\}^infty\; \{T\_n(x)\; over\; n!\}\; t^n=e^\{xleft(e^t-1\; ight)\}.$