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

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

(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)

See also Actuarial Polynomial ,

Bell Number ,

Complementary
Bell Number ,

Dobiński's Formula ,

Idempotent Number ,

Lah
Number ,

Sheffer Sequence ,

Stirling
Number of the Second Kind
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References Bell, E. T. "Exponential Polynomials." Ann. Math. 35 , 258-277, 1934. Comtet, L. Advanced
Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht,
Netherlands: Reidel, p. 133, 1974. Riordan, J. An
Introduction to Combinatorial Analysis. New York: Wiley, pp. pp. 35-38,
49, and 142, 1980. Roman, S. "The Exponential Polynomials"
and "The Bell Polynomials." §4.1.3 and §4.1.8 in The
Umbral Calculus. New York: Academic Press, pp. 63-67 and 82-87, 1984. Sloane,
N. J. A. Sequence A106800 in "The
On-Line Encyclopedia of Integer Sequences." Referenced on Wolfram|Alpha Bell Polynomial
Cite this as:
Weisstein, Eric W. "Bell Polynomial."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/BellPolynomial.html

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