Bell Polynomial

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

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

(1)

where is the univariate Bell polynomial, the multivariate complete Bell polynomial, and the multivariate partial Bell polynomial. All three families are called Bell polynomials. A generalized Bell polynomial expresses derivatives of a function composition of any finite number of functions. An alternative convention defines ordinary partial Bell polynomials using ordinary generating functions, whose terms contain rather than the used in exponential generating functions. This changes the coefficients of the polynomials.

A univariate 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

			(2)
			(3)

These univariate Bell polynomials generalize the exponential function.

The univariate Bell polynomials should not be confused with Bernoulli polynomials, which are also commonly denoted .

The univariate Bell polynomials are implemented in the Wolfram Language as BellB[n, x].

The first few univariate Bell polynomials are

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)

(OEIS A106800).

forms the associated Sheffer sequence for

(11)

so these polynomials have the exponential generating function

(12)

An explicit sum for is

(13)

with the convention when . A recurrence relation is

(14)

with , where is a binomial coefficient.

The univariate Bell polynomials have the explicit formula

(15)

where is a Stirling number of the second kind.

A binomial sum is given by

(16)

where is a binomial coefficient.

For , the derivative of is given by

(17)

so satisfies the recurrence equation

(18)