Touchard polynomials

{{for|a different family of polynomials Q_n occasionally called Touchard polynomials|Bateman polynomials}}

The Touchard polynomials, studied by {{harvs|txt|authorlink=Jacques Touchard|first=Jacques|last= Touchard|year=1939}},{{Citation | last1=Touchard | first1=Jacques | title=Sur les cycles des substitutions | doi=10.1007/BF02547349 | mr=1555449 | year=1939 | journal=Acta Mathematica | issn=0001-5962 | volume=70 | issue=1 | pages=243–297| doi-access=free }} also called the exponential polynomials or Bell polynomials, 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.{{cite book|last=Roman|first=Steven|title=The Umbral Calculus|year=1984|publisher=Dover|isbn=0-486-44139-3}}{{cite journal|last=Boyadzhiev|first=Khristo N.|title=Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals |arxiv=0909.0979|doi=10.1155/2009/168672|volume=2009|journal=Abstract and Applied Analysis|date=2009 |pages=1–18|bibcode=2009AbApA2009....1B |doi-access=free }}{{cite web|last=Brendt|first=Bruce C|title=RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU|url=http://www.math.uiuc.edu/~berndt/articles/gravesnatching.pdf|accessdate=23 November 2013}}{{MathWorld|urlname=BellPolynomial|title=Bell Polynomial}}

The first few Touchard polynomials are

: $T_1(x)=x,$

: $T_2(x)=x^2+x,$

: $T_3(x)=x^3+3x^2+x,$

: $T_4(x)=x^4+6x^3+7x^2+x,$

: $T_5(x)=x^5+10x^4+25x^3+15x^2+x.$

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:

: $T_n(1)=B_n.$

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:

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

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.

A generalization of both this formula and the definition, is a generalization of Spivey's formula{{Cite web |title=Implications of Spivey's Bell Number Formula |url=https://cs.uwaterloo.ca/journals/JIS/VOL11/Gould/gould35.html |access-date=2023-05-28 |website=cs.uwaterloo.ca}}

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

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

: $T_n(\lambda+\mu)=\left(T(\lambda)+T(\mu) \right)^n,$ {{clarification needed|date=April 2024}}

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

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:

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

= Zeroes =

All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967.{{Cite journal

| last = Harper | first = L. H.

| title = Stirling behavior is asymptotically normal

| year = 1967

| volume = 38

| issue = 2

| pages = 410–414

| journal = The Annals of Mathematical Statistics

| doi=10.1214/aoms/1177698956

| doi-access = free

}}

The absolute value of the leftmost zero is bounded from above by{{Cite journal

| last1 = Mező | first1 = István

| last2 = Corcino | first2 = Roberto B.

| title = The estimation of the zeros of the Bell and r-Bell polynomials

| year = 2015

| volume = 250

| pages = 727–732

| journal = Applied Mathematics and Computation

| doi=10.1016/j.amc.2014.10.058

}}

: $\frac1n\binom{n}{2}+\frac{n-1}{n}\sqrt{\binom{n}{2}^2-\frac{2n}{n-1}\left(\binom{n}{3}+3\binom{n}{4}\right)},$

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

The Mahler measure $M(T_n)$ of the Touchard polynomials can be estimated as follows:{{cite web|last1=István|first1=Mező|title=On the Mahler measure of the Bell polynomials|url=https://sites.google.com/site/istvanmezo81/others|accessdate=7 November 2017}}

: $\frac{\lbrace\textstyle{n\atop \Omega_n}\rbrace}{\binom{n}{\Omega_n}}\le M(T_n)\le\sqrt{n+1}\left\{{n\atop K_n}\right\},$

where $\Omega_n$ and $K_n$ are the smallest of the maximum two k indices such that

$\lbrace\textstyle{n\atop k}\rbrace/\binom{n}{k}$ and $\lbrace\textstyle{n\atop k}\rbrace$

are maximal, respectively.

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

References

Category:Polynomials