Editing Touchard polynomials (section)

{{Use American English|date = March 2019}}
{{Short description|Sequence of polynomials}}
{{for|a different family of polynomials Q<sub>n</sub> occasionally called Touchard polynomials|Bateman polynomials}}
{{distinguish|Bell polynomials}}

[[File:Touchard_Polynomials.png | thumb | right | Touchard Polynomials]]
The '''Touchard polynomials''', studied by {{harvs|txt|authorlink=Jacques Touchard|first=Jacques|last= Touchard|year=1939}},<ref>{{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 }}</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 [[partition of a set|partitions of a set]] of size ''n'' into ''k'' disjoint non-empty subsets.<ref name=Roman>{{cite book|last=Roman|first=Steven|title=The Umbral Calculus|year=1984|publisher=Dover|isbn=0-486-44139-3}}</ref><ref>{{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 }}</ref><ref>{{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}}</ref><ref>{{MathWorld|urlname=BellPolynomial|title=Bell Polynomial}}</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>