Touchard polynomials

The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by

${\displaystyle T_{n}(x)=\sum _{k=0}^{n}S(n,k)x^{k}=\sum _{k=0}^{n}\left\{{n \atop k}\right\}x^{k},}$

where ${\displaystyle S(n,k)=\left\{{n \atop k}\right\}}$ is 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.^[1][2][3][4]

The first few Touchard polynomials are

${\displaystyle T_{1}(x)=x,}$

${\displaystyle T_{2}(x)=x^{2}+x,}$

${\displaystyle T_{3}(x)=x^{3}+3x^{2}+x,}$

${\displaystyle T_{4}(x)=x^{4}+6x^{3}+7x^{2}+x,}$

${\displaystyle T_{5}(x)=x^{5}+10x^{4}+25x^{3}+15x^{2}+x.}$

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:

${\displaystyle T_{n}(1)=B_{n}.}$

If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(Xⁿ) = T_n(λ), leading to the definition:

${\displaystyle T_{n}(x)=e^{-x}\sum _{k=0}^{\infty }{\frac {x^{k}k^{n}}{k!}}.}$

Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities:

${\displaystyle T_{n}(\lambda +\mu )=\sum _{k=0}^{n}{n \choose k}T_{k}(\lambda )T_{n-k}(\mu ).}$

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:

${\displaystyle T_{n}\left(e^{x}\right)=e^{-e^{x}}{\frac {d^{n}}{dx^{n}}}\;e^{e^{x}}.}$

The Touchard polynomials satisfy the recurrence relation

${\displaystyle T_{n+1}(x)=x\left(1+{\frac {d}{dx}}\right)T_{n}(x)}$

and

${\displaystyle T_{n+1}(x)=x\sum _{k=0}^{n}{n \choose k}T_{k}(x).}$

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^[5]

$${\displaystyle T_{n+m}(x)=\sum _{k=0}^{n}\left\{{n \atop k}\right\}x^{k}\sum _{j=0}^{m}{\binom {m}{j}}k^{m-j}T_{j}(x)}$$

Using the umbral notation Tⁿ(x)=T_n(x), these formulas become:

${\displaystyle T_{n}(\lambda +\mu )=\left(T(\lambda )+T(\mu )\right)^{n},}$^{[clarification needed]}

${\displaystyle T_{n+1}(x)=x\left(1+T(x)\right)^{n}.}$

The generating function of the Touchard polynomials is

${\displaystyle \sum _{n=0}^{\infty }{T_{n}(x) \over n!}t^{n}=e^{x\left(e^{t}-1\right)},}$

which corresponds to the generating function of Stirling numbers of the second kind.

Touchard polynomials have contour integral representation:

${\displaystyle T_{n}(x)={\frac {n!}{2\pi i}}\oint {\frac {e^{x({e^{t}}-1)}}{t^{n+1}}}\,dt.}$

All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967.^[6]

The absolute value of the leftmost zero is bounded from above by^[7]

${\displaystyle {\frac {1}{n}}{\binom {n}{2}}+{\frac {n-1}{n}}{\sqrt {{\binom {n}{2}}^{2}-{\frac {2n}{n-1}}\left({\binom {n}{3}}+3{\binom {n}{4}}\right)}},}$

although it is conjectured that the leftmost zero grows linearly with the index n.

The Mahler measure ${\displaystyle M(T_{n})}$ of the Touchard polynomials can be estimated as follows:^[8]

${\displaystyle {\frac {\lbrace \textstyle {n \atop \Omega _{n}}\rbrace }{\binom {n}{\Omega _{n}}}}\leq M(T_{n})\leq {\sqrt {n+1}}\left\{{n \atop K_{n}}\right\},}$

where ${\displaystyle \Omega _{n}}$ and ${\displaystyle K_{n}}$ are the smallest of the maximum two k indices such that ${\displaystyle \lbrace \textstyle {n \atop k}\rbrace /{\binom {n}{k}}}$ and ${\displaystyle \lbrace \textstyle {n \atop k}\rbrace }$ are maximal, respectively.

• Complete Bell polynomial ${\displaystyle B_{n}(x_{1},x_{2},\dots ,x_{n})}$ may be viewed as a multivariate generalization of Touchard polynomial ${\displaystyle T_{n}(x)}$, since ${\displaystyle T_{n}(x)=B_{n}(x,x,\dots ,x).}$
• The Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order:

  ${\displaystyle T_{n}(x)={\frac {n!}{\pi }}\int _{0}^{\pi }e^{x{\bigl (}e^{\cos(\theta )}\cos(\sin(\theta ))-1{\bigr )}}\cos {\bigl (}xe^{\cos(\theta )}\sin(\sin(\theta ))-n\theta {\bigr )}\,d\theta \,.}$

• Bell polynomials

• Touchard, Jacques (1939), "Sur les cycles des substitutions", Acta Mathematica, 70 (1): 243–297, doi:10.1007/BF02547349, ISSN 0001-5962, MR 1555449