Bell Polynomial

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)