BellB
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
- The Bell polynomials satisfy the generating function relation
.
- The Bell numbers are given by
.
- For certain special arguments, BellB automatically evaluates to exact values.
- BellB can be evaluated to arbitrary numerical precision.
- BellB automatically threads over lists.
Background & Context
- BellB is a mathematical function that returns a Bell number or polynomial. In particular, BellB[n,x] returns the
Bell polynomial
and BellB[n] returns the
Bell number
. Bell polynomials can be determined from the exponential generating function
. The Bell numbers also satisfy the recurrence relation
. The first few Bell polynomials
are
, while the first few Bell numbers
are
.
- The Bell polynomial is also called an exponential polynomial or, more explicitly, the "complete exponential Bell polynomial" and is sometimes denoted
. Bell polynomials are named after mathematician and math expositor Eric Temple Bell, who wrote about them in 1934.
- The polynomial
has the interpretation that if there are
partitions of
into
parts, then
. Furthermore, if there are
total partitions of
, then
. For example, the set
having
elements can be partitioned into
parts
ways
,
part
way (
),
parts
ways (
,
and
), and
parts
way (
), giving
. Since there are five total ways to partition
,
.
- The Bell polynomial and number are a special case of the BellY function, with
and
. Letting
denote the Stirling number of the second kind, returned by StirlingS2,
.
Examples
open allclose allScope (4)
The precision of the output tracks the precision of the input:
BellB threads element-wise over lists:
TraditionalForm formatting:
Generalizations & Extensions (1)
BellB can be applied to a power series:
Applications (2)
Properties & Relations (8)
Sum can give results involving BellB:
Moment of PoissonDistribution is given by BellB polynomial in its mean :
Use FullSimplify to simplify expressions involving BellB:
Compute Bell numbers directly from set partitions :
Use IntegerPartitions to directly sum over terms that satisfy the constraints on indices:
Compare to the built-in function values:
Compute Bell numbers using generalized Bell polynomials:
FindSequenceFunction can recognize the BellB sequence:
The exponential generating function for BellB:
Possible Issues (1)
The first argument of BellB must be a non-negative integer:

Text
Wolfram Research (2007), BellB, Wolfram Language function, https://reference.wolfram.com/language/ref/BellB.html.
CMS
Wolfram Language. 2007. "BellB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BellB.html.
APA
Wolfram Language. (2007). BellB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BellB.html