Bell Polynomial

The term "Bell polynomial" refers to univariate exponential polynomials and to related multivariate polynomials.

A Bell polynomial , also called an exponential polynomial and denoted (Bell 1934, Roman 1984, pp. 63-67) is a polynomial, for a nonnegative integer , that generalizes the Bell number and complementary Bell number such that

			(1)
			(2)

These Bell polynomials 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 these polynomials have the exponential generating function

(11)

An explicit sum for is

(12)

with the convention when . A recurrence relation is

(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 binomial sum is given by

(15)

where is a binomial coefficient.

For , the derivative of is given by

(16)

so satisfies the recurrence equation

(17)

The diagram above shows all 15 set partitions of four objects, grouped by the number of blocks. Each set block of size contributes a factor , and the row labels give the partial Bell polynomials . Their sum is the complete Bell polynomial .

The multivariate family consists of partial Bell polynomials and their complete Bell polynomial sums. In the exponential convention, the univariate Bell polynomial is the specialization

(18)

where is the complete Bell polynomial and the partial Bell polynomial. This distinguishes the univariate from the multivariate and , although all are called Bell polynomials. The partial Bell polynomials also have an ordinary normalization with different factorial weights.