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

$T_n(x)=\sum_{k=0}^n S(n,k)x^k=\sum_{k=0}^n \left\{ {n \atop k} \right\}x^k,$

where $S(n,k)=\left\{ {n \atop k} \right\}$ 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:

$T_n(1)=B_n.$

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:

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

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:

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

The Touchard polynomials satisfy the recurrence relation

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

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.

Using the umbral notation Tⁿ(x)=T_n(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:

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

Generalizations

• Complete Bell polynomial $B_n(x_1,x_2,\dots,x_n)$ may be viewed as a multivariate generalization of Touchard polynomial $T_n(x)$, since $T_n(x) = B_n(x,x,\dots,x).$
• The Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order:

$T_n(x)=\frac{n!}{\pi} \int^{\pi}_0 e^{x \bigl(e^{\cos(\theta)} \cos(\sin(\theta))-1 \bigr)} \cos \bigl(x e^{\cos(\theta)} \sin(\sin(\theta)) -n\theta\bigr) \, \mathrm{d}\theta$

See also

• Bell polynomials

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" (PDF). arxiv. Retrieved 23 November 2013.
3. Brendt, Bruce C. "RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU" (PDF). Retrieved 23 November 2013.
4. Weisstein, Eric W., "Bell Polynomial", MathWorld.

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