# Touchard polynomials

The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials[1][2][3] or Bell polynomials,[4] 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.

## 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:

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

If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(Xn) = Tn(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:

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

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.

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

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

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:

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

The Touchard polynomials have only real and negative roots. This fact was proven by L. H. Harper in 1967.[5] The leftmost root is bounded from below (in absolute value) by[6]

${\displaystyle {\frac {1}{n}}{\binom {n}{2}}+{\frac {n-1}{n}}{\sqrt {{\binom {n}{2}}^{2}-{\frac {2n}{n-1}}\left({\binom {n}{3}}+3{\binom {n}{4}}\right)}},}$ although it is believed by the same authors that the leftmost root grows linearly with the index n.

## Generalizations

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

## References

1. ^ Roman, Steven (1984). The Umbral Calculus. Dover. ISBN 0-486-44139-3.
2. ^ Boyadzhiev, Khristo N. "Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals". arXiv:.
3. ^ Brendt, Bruce C. "RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU" (PDF). Retrieved 23 November 2013.
4. ^
5. ^ Harper, L. H. (1967). "Stirling behavior is asymptotically normal". The Annals of Mathematical Statistics. 38 (2): 410–414. doi:10.1214/aoms/1177698956.
6. ^ Mező, István; Corcino, Roberto B. (2015). "The estimation of the zeros of the Bell and r-Bell polynomials". Applied Mathematics and Computation. 250: 727–732. doi:10.1016/j.amc.2014.10.058.