Editing Bell polynomials (section)

==Applications==

===Faà di Bruno's formula===
{{main|Faà di Bruno's formula}}

[[Faà di Bruno's formula]] may be stated in terms of Bell polynomials as follows:

:<math>{d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x)) B_{n,k} \left(g'(x),g''(x), \dots, g^{(n-k+1)}(x)\right).</math>

Similarly, a power-series version of Faà di Bruno's formula may be stated using Bell polynomials as follows.  Suppose

:<math>f(x)=\sum_{n=1}^\infty {a_n \over n!} x^n \qquad \text{and} \qquad g(x) = \sum_{n=0}^\infty {b_n \over n!} x^n.</math>

Then

:<math>g(f(x)) = \sum_{n=1}^\infty
\frac{\sum_{k=0}^n b_k B_{n,k}(a_1,\dots,a_{n-k+1})}{n!} x^n.</math>

In particular, the complete Bell polynomials appear in the exponential of a [[formal power series]]:

:<math>\exp\left(\sum_{i=1}^\infty {a_i \over i!} x^i \right)
= \sum_{n=0}^\infty {B_n(a_1,\dots,a_n) \over n!} x^n,</math>
which also represents the [[exponential generating function]] of the complete Bell polynomials on a fixed sequence of arguments <math>a_1, a_2, \dots</math>.

===Reversion of series===
{{main|Lagrange inversion theorem}}

Let two functions ''f'' and ''g'' be expressed in formal [[power series]] as

:<math>f(w) = \sum_{k=0}^\infty f_k \frac{w^k}{k!}, \qquad \text{and}  \qquad  g(z) = \sum_{k=0}^\infty g_k \frac{z^k}{k!},</math>

such that ''g'' is the compositional inverse of ''f'' defined by ''g''(''f''(''w'')) = ''w'' or ''f''(''g''(''z'')) = ''z''. If ''f''<sub>0</sub> = 0 and ''f''<sub>1</sub> ≠ 0, then an explicit form of the coefficients of the inverse can be given in term of Bell polynomials as{{sfn|Charalambides|2002|p=437|loc=Eqn (11.43)}}

:<math> g_n = \frac{1}{f_1^n} \sum_{k=1}^{n-1} (-1)^k n^{\bar{k}} B_{n-1,k}(\hat{f}_1,\hat{f}_2,\ldots,\hat{f}_{n-k}), \qquad n \geq 2, </math>

with <math> \hat{f}_k = \frac{f_{k+1}}{(k+1)f_{1}},</math> and <math>n^{\bar{k}} = n(n+1)\cdots (n+k-1) </math> is the rising factorial, and <math>g_1 = \frac{1}{f_{1}}. </math>

===Asymptotic expansion of Laplace-type integrals===
Consider the integral of the form

:<math>I(\lambda) = \int_a^b e^{-\lambda f(x)} g(x) \, \mathrm{d}x, </math>

where (''a'',''b'') is a real (finite or infinite) interval, λ is a large positive parameter and the functions ''f'' and ''g'' are continuous. Let ''f'' have a single minimum in [''a'',''b''] which occurs at ''x''&nbsp;=&nbsp;''a''. Assume that as ''x''&nbsp;→&nbsp;''a''<sup>+</sup>,

:<math> f(x) \sim f(a) + \sum_{k=0}^\infty a_k (x-a)^{k+\alpha}, </math>
:<math> g(x) \sim \sum_{k=0}^\infty b_k (x-a)^{k+\beta-1}, </math>

with ''α'' > 0, Re(''β'') > 0; and that the expansion of ''f'' can be term wise differentiated. Then, Laplace–Erdelyi theorem states that the asymptotic expansion of the integral ''I''(''λ'') is given by

:<math> I(\lambda) \sim e^{-\lambda f(a)} \sum_{n=0}^\infty \Gamma \Big(\frac{n+\beta}{\alpha} \Big) \frac{c_n}{\lambda^{(n+\beta)/\alpha}} \qquad \text{as} \quad \lambda \rightarrow \infty, </math>

where the coefficients ''c<sub>n</sub>'' are expressible in terms of ''a<sub>n</sub>'' and ''b<sub>n</sub>'' using partial ''ordinary'' Bell polynomials, as given by Campbell–Froman–Walles–Wojdylo formula:

:<math> c_n = \frac{1}{\alpha a_0^{(n+\beta)/\alpha}} \sum_{k=0}^n b_{n-k} \sum_{j=0}^k \binom{-\frac{n+\beta}{\alpha}}{j} \frac{1}{a_0^j} \hat{B}_{k,j}(a_1,a_2,\ldots,a_{k-j+1}). </math>

===Symmetric polynomials=== 
{{main|Newton's identities}}
The [[elementary symmetric polynomial]] <math>e_n</math> and the [[power sum symmetric polynomial]] <math>p_n</math> can be related to each other using Bell polynomials as:
: <math>
\begin{align}
e_n & = \frac{1}{n!}\; B_{n}(p_1, -1! p_2, 2! p_3, -3! p_4, \ldots, (-1)^{n-1}(n-1)! p_n ) \\
& = \frac{(-1)^n}{n!}\; B_{n}(-p_1, -1! p_2, -2! p_3, -3! p_4, \ldots, -(n-1)! p_n ),
\end{align}
</math>
: <math>
\begin{align}
p_n & = \frac{(-1)^{n-1}}{(n-1)!} \sum_{k=1}^n (-1)^{k-1} (k-1)!\; B_{n,k}(e_1,2! e_2, 3! e_3,\ldots,(n-k+1)! e_{n-k+1}) \\
& = (-1)^n\; n\; \sum_{k=1}^n \frac{1}{k} \; \hat{B}_{n,k}(-e_1,\dots,-e_{n-k+1}).
\end{align}
</math>


These formulae allow one to express the coefficients of monic polynomials in terms of the Bell polynomials of its zeroes. For instance, together with [[Cayley–Hamilton theorem]] they lead to expression of the determinant of a ''n'' × ''n'' square matrix ''A'' in terms of the traces of its powers: 
: <math> \det (A) = \frac{(-1)^{n}}{n!} B_n(s_1, s_2, \ldots, s_n), ~\qquad \text{where }  s_k = - (k - 1)! \operatorname{tr}(A^k).</math>

===Cycle index of symmetric groups===
{{main|Cycle index}}

The [[cycle index]] of the [[symmetric group]] <math>S_n</math> can be expressed in terms of complete Bell polynomials as follows:
:<math> Z(S_n) = \frac{B_n(0!\,a_1, 1!\,a_2, \dots, (n-1)!\,a_n)}{n!}.</math>

===Moments and cumulants===

The sum

:<math>\mu_n' = B_n(\kappa_1,\dots,\kappa_n)=\sum_{k=1}^n B_{n,k}(\kappa_1,\dots,\kappa_{n-k+1})</math>

is the ''n''th raw [[moment (mathematics)|moment]] of a [[probability distribution]] whose first ''n'' [[cumulant]]s are ''κ''<sub>1</sub>, ..., ''κ''<sub>''n''</sub>.  In other words, the ''n''th moment is the ''n''th complete Bell polynomial evaluated at the first ''n'' cumulants. Likewise, the ''n''th cumulant can be given in terms of the moments as

:<math>\kappa_n = \sum_{k=1}^n (-1)^{k-1} (k-1)! B_{n,k}(\mu'_1,\ldots,\mu'_{n-k+1}).</math>

===Hermite polynomials===
{{main|Hermite polynomials}}

[[Hermite polynomials]] can be expressed in terms of Bell polynomials as

:<math>\operatorname{He}_n(x) = B_n(x,-1,0,\ldots,0),</math>

where ''x''<sub>''i''</sub> = 0 for all ''i'' > 2; thus allowing for a combinatorial interpretation of the coefficients of the Hermite polynomials. This can be seen by comparing the generating function of the Hermite polynomials

:<math>\exp \left(xt-\frac{t^2}{2} \right) = \sum_{n=0}^\infty \operatorname{He}_n(x) \frac {t^n}{n!}</math>

with that of Bell polynomials.

===Representation of polynomial sequences of binomial type===

For any sequence ''a''<sub>1</sub>, ''a''<sub>2</sub>, …, ''a''<sub>''n''</sub> of scalars, let

:<math>p_n(x)= B_n(a_1 x, \ldots, a_n x) = \sum_{k=1}^n B_{n,k}(a_1,\dots,a_{n-k+1}) x^k.</math>

Then this polynomial sequence is of [[binomial type]], i.e. it satisfies the binomial identity

:<math>p_n(x+y)=\sum_{k=0}^n {n \choose k} p_k(x) p_{n-k}(y).</math>
:'''Example:''' For ''a''<sub>1</sub> = … = ''a''<sub>''n''</sub> = 1, the polynomials <math>p_n(x)</math> represent [[Touchard polynomials]].

More generally, we have this result:

:'''Theorem:''' All polynomial sequences of binomial type are of this form.

If we define a formal power series

:<math>h(x)=\sum_{k=1}^\infty {a_k \over k!} x^k,</math>

then for all ''n'',

:<math>h^{-1}\left( {d \over dx}\right) p_n(x) = n p_{n-1}(x).</math>