Editing Touchard polynomials (section)

=== 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.