Touchard polynomials

File:Touchard Polynomials.png

Touchard Polynomials

The Touchard polynomials, studied by Template:Harvs,<ref>Template:Citation</ref> also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by

<math>T_n(x)=\sum_{k=0}^n S(n,k)x^k=\sum_{k=0}^n

\left\{ {n \atop k} \right\}x^k,</math>

where <math>S(n,k)=\left\{ {n \atop k} \right\}</math> 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.<ref name=Roman>Template:Cite book</ref><ref>Template:Cite journal</ref><ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref><ref>{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web |_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:BellPolynomial%7CBellPolynomial.html}} |title = Bell Polynomial |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}</ref>

The first few Touchard polynomials are

<math>T_1(x)=x,</math>

<math>T_2(x)=x^2+x,</math>

<math>T_3(x)=x^3+3x^2+x,</math>

<math>T_4(x)=x^4+6x^3+7x^2+x,</math>

<math>T_5(x)=x^5+10x^4+25x^3+15x^2+x.</math>

Properties

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:

<math>T_n(1)=B_n.</math>

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:

<math>T_{n}(x)=e^{-x}\sum_{k=0}^\infty \frac {x^k k^n} {k!}.</math>

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

<math>T_n(\lambda+\mu)=\sum_{k=0}^n {n \choose k} T_k(\lambda) T_{n-k}(\mu).</math>

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:

<math>T_n \left(e^x \right) = e^{-e^x} \frac{d^n}{dx^n}\;e^{e^x}.</math>

The Touchard polynomials satisfy the recurrence relation

<math>T_{n+1}(x)=x \left(1+\frac{d}{dx} \right)T_{n}(x)</math>

and

<math>T_{n+1}(x)=x\sum_{k=0}^n{n \choose k}T_k(x).</math>

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<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math display="block">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)</math>

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

<math>T_n(\lambda+\mu)=\left(T(\lambda)+T(\mu) \right)^n,</math>Template:Clarification needed

<math>T_{n+1}(x)=x \left(1+T(x) \right)^n.</math>

The generating function of the Touchard polynomials is

<math>\sum_{n=0}^\infty {T_n(x) \over n!} t^n=e^{x\left(e^t-1\right)},</math>

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

Touchard polynomials have contour integral representation:

<math>T_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x({e^t}-1)}}{t^{n+1}}\,dt.</math>

Zeroes

All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967.<ref name = 'Harper'>Template:Cite journal</ref>

The absolute value of the leftmost zero is bounded from above by<ref name = 'MC'>Template:Cite journal</ref>

<math>\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)},</math>

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

The Mahler measure <math>M(T_n)</math> of the Touchard polynomials can be estimated as follows:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

<math>

\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\}, </math> where <math>\Omega_n</math> and <math>K_n</math> are the smallest of the maximum two k indices such that <math>\lbrace\textstyle{n\atop k}\rbrace/\binom{n}{k}</math> and <math>\lbrace\textstyle{n\atop k}\rbrace</math> are maximal, respectively.

Generalizations

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

  <math>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) \, d\theta\, .</math>

See also

• Bell polynomials