Touchard polynomials Guide, Meaning , Facts, Information and Description
The
Touchard 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
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
This is an Article on Touchard polynomials. Page Contains Information, Facts Details or Explanation Guide About Touchard polynomials
