Touchard polynomials
sequence of polynomials
The Touchard polynomials, studied by Jacques Touchard (1956), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by
T
n
(
x
)
=
∑
k
=
0
n
S
(
n
,
k
)
x
k
=
∑
k
=
0
n
{
n
k
}
x
k
,
{\displaystyle 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
)
=
{
n
k
}
{\displaystyle 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.
The first few Touchard polynomials are
T
1
(
x
)
=
x
,
{\displaystyle T_{1}(x)=x,}
T
2
(
x
)
=
x
2
+
x
,
{\displaystyle T_{2}(x)=x^{2}+x,}
T
3
(
x
)
=
x
3
+
3
x
2
+
x
,
{\displaystyle T_{3}(x)=x^{3}+3x^{2}+x,}
T
4
(
x
)
=
x
4
+
6
x
3
+
7
x
2
+
x
,
{\displaystyle T_{4}(x)=x^{4}+6x^{3}+7x^{2}+x,}
T
5
(
x
)
=
x
5
+
10
x
4
+
25
x
3
+
15
x
2
+
x
.
{\displaystyle 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
.
{\displaystyle 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) = Tn(λ), leading to the definition:
T
n
(
x
)
=
e
−
x
∑
k
=
0
∞
x
k
k
n
k
!
.
{\displaystyle 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
(
λ
+
μ
)
=
∑
k
=
0
n
(
n
k
)
T
k
(
λ
)
T
n
−
k
(
μ
)
.
{\displaystyle 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
(
e
x
)
=
e
−
e
x
d
n
d
x
n
e
e
x
.
{\displaystyle 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
(
T
n
(
x
)
+
T
n
′
(
x
)
)
{\displaystyle T_{n+1}(x)=x\left(T_{n}(x)+T'_{n}(x)\right)}
and
T
n
+
1
(
x
)
=
x
∑
k
=
0
n
(
n
k
)
T
k
(
x
)
.
{\displaystyle 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.
“Touchard polynomials” enters the record as sequence of polynomials. Crown Archives preserves that source wording while asking what Touchard, polynomials and sequence can confirm, complicate or overturn.
Why this record matters
“Touchard polynomials” is worth following because a concise public description often conceals a longer documentary argument. Here, Touchard, polynomials and sequence provides the most credible route into that argument.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Sep 7, 2026. The linked authority identifier is Q3820608. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1956.
Overview language is designed for orientation and should not be treated as a substitute for the evidence cited beneath it. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Touchard polynomials”, its source revision and the description used here.
- Expand the search: follow Touchard polynomials primary sources, Touchard polynomials archive and Touchard research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Touchard polynomials”?
- Which institution is responsible for the underlying evidence?
- What terminology or title could unlock a more precise catalogue search?
Search terms from this dossier
This entry incorporates text from “Touchard polynomials” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.