There are two kinds of Bell polynomials.
A Bell polynomial , also called an exponential polynomial and denoted (Bell 1934, Roman 1984, pp. 63-67) is a polynomial that generalizes the Bell number and complementary Bell number such that
(1)
| |||
(2)
|
These Bell polynomial generalize the exponential function.
Bell polynomials should not be confused with Bernoulli polynomials, which are also commonly denoted .
Bell polynomials are implemented in the Wolfram Language as BellB[n, x].
The first few Bell polynomials are
(3)
| |||
(4)
| |||
(5)
| |||
(6)
| |||
(7)
| |||
(8)
| |||
(9)
|
(OEIS A106800).
forms the associated Sheffer sequence for
(10)
|
so the polynomials have that exponential generating function
(11)
|
Additional generating functions for are given by
(12)
|
or
(13)
|
with , where is a binomial coefficient.
The Bell polynomials have the explicit formula
(14)
|
where is a Stirling number of the second kind.
A beautiful binomial sum is given by
(15)
|
where is a binomial coefficient.
The derivative of is given by
(16)
|
so satisfies the recurrence equation
(17)
|
The second kind of Bell polynomials are defined by
(18)
|
They have generating function
(19)
|