Touchard polynomials

The Touchard polynomials, studied by , also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by
where formula_2is a Stirling number of the second kind, i.e., the number of partitions of a set of size "n" into "k" disjoint non-empty subsets.
Properties.
Basic properties.
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 a Poisson distribution with expected value λ, then its "n"th moment is E("X""n") = "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 constitute the only polynomial sequence of binomial type with the coefficient of "x" equal 1 in every polynomial.
The Touchard polynomials satisfy the Rodrigues-like formula:
The Touchard polynomials satisfy the recurrence relation
and
In the case "x" = 1, this reduces to the recurrence formula for the Bell numbers.
Using the umbral notation "T""n"("x")="T""n"("x"), these formulas become:
The generating function of the Touchard polynomials is
which corresponds to the generating function of Stirling numbers of the second kind.
Touchard polynomials have contour integral representation:
Zeroes.
All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967. 
The smallest zero is bounded from below (in absolute value) by
although it is conjectured that the smallest zero grows linearly with the index "n".
The Mahler measure formula_14of the Touchard polynomials can be estimated as follows:
\frac{\lbrace\textstyle{n\atop \Omega_n}\rbrace}{\binom{n}{\Omega_n}}\le M(T_n)\le\sqrt{n+1}\left\