Touchard polynomials

Touchard polynomials

The Touchard polynomials, named after Jacques Touchard, also called the exponential polynomials, comprise a polynomial sequence of binomial type defined by

:T_n(x)=sum_{k=1}^n S(n,k)x^k=sum_{k=1}^nleft{egin{matrix} n \ k end{matrix} ight}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 "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)}.


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