- Touchard polynomials
The Touchard polynomials, named after
Jacques Touchard , also called the exponential polynomials, comprise apolynomial sequence ofbinomial 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 byDonald 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"::
If "X" is a
random variable with aPoisson 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 thispolynomial sequence is ofbinomial 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:
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