Editing Bell polynomials (section)

===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>