Editing Touchard polynomials (section)

== Properties ==

=== Basic properties ===
The value at 1 of the ''n''th Touchard polynomial is the ''n''th [[Bell numbers|Bell number]], i.e., the number of [[partition of a set|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 ''n''th moment is E(''X''<sup>''n''</sup>) = ''T''<sub>''n''</sub>(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>{{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}}</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 calculus|umbral notation]] ''T''<sup>''n''</sup>(''x'')=''T''<sub>''n''</sub>(''x''), these formulas become:
:<math>T_n(\lambda+\mu)=\left(T(\lambda)+T(\mu) \right)^n,</math>{{clarification needed|date=April 2024}}
:<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 [[Stirling numbers of the second kind#Generating functions|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'>{{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
}}</ref> 

The absolute value of the leftmost zero is bounded from above by<ref name = 'MC'>{{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
}}</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>{{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}}</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.