Touchard polynomials

For a different family of polynomials Q_n occasionally called Touchard polynomials, see Bateman polynomials.

Not to be confused with Bell 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)}},

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.

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 _{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

↑ Roman, Steven (1984). The Umbral Calculus. Dover. ISBN 0-486-44139-3.
↑ Boyadzhiev, Khristo N. "Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals". arXiv:0909.0979.
↑ Brendt, Bruce C. "RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU" (PDF). Retrieved 23 November 2013.
↑ 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

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