Basic properties 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: :T_n(1)=B_n. If
X is a
random variable with a
Poisson distribution with
expected value λ, then its
nth moment is E(
Xn) =
Tn(λ), leading to the definition: :T_{n}(x)=e^{-x}\sum_{k=0}^\infty \frac {x^k k^n} {k!}. 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 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: :T_n \left(e^x \right) = e^{-e^x} \frac{d^n}{dx^n}\;e^{e^x}. The Touchard polynomials satisfy the
recurrence relation :T_{n+1}(x)=x \left(T_{n}(x) + T'_{n}(x)\right) and :T_{n+1}(x)=x\sum_{k=0}^n{n \choose k}T_k(x). In the case
x = 1, this reduces to the recurrence formula for the
Bell numbers. A generalization of both this formula and the definition, is a generalization of Spivey's formula T_{n+m}(x) = \sum_{k=0}^n \left\{ {n \atop k} \right\} x^k \sum_{j=0}^m \binom{m}{j} k^{m-j} T_j(x) Using the
umbral notation Tn(
x)=
Tn(
x), these formulas become: :T_n(\lambda+\mu)=\left(T(\lambda)+T(\mu) \right)^n, :T_{n+1}(x)=x \left(1+T(x) \right)^n. The
generating function of the Touchard polynomials is :\sum_{n=0}^\infty {T_n(x) \over n!} t^n=e^{x\left(e^t-1\right)}, which corresponds to the
generating function of Stirling numbers of the second kind. Touchard polynomials have
contour integral representation: :T_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x({e^t}-1)}}{t^{n+1}}\,dt.
Zeroes All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967. The absolute value of the leftmost zero is bounded from above by :\frac1n\binom{n}{2}+\frac{n-1}{n}\sqrt{\binom{n}{2}^2-\frac{2n}{n-1}\left(\binom{n}{3}+3\binom{n}{4}\right)}, although it is conjectured that the leftmost zero grows linearly with the index
n. The
Mahler measure M(T_n) of 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\{{n\atop K_n}\right\}, where \Omega_n and K_n are the smallest of the maximum two
k indices such that \lbrace\textstyle{n\atop k}\rbrace/\binom{n}{k} and \lbrace\textstyle{n\atop k}\rbrace are maximal, respectively. == Generalizations ==