# BellB

BellB[n]

gives the Bell number .

BellB[n,x]

gives the Bell polynomial .

# 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 all

## Basic Examples(2)

The tenth Bell number:

The fifth Bell polynomial:

## Scope(5)

Evaluate numerically:

The precision of the output tracks the precision of the input:

BellB can be applied to a power series:

## Applications(4)

BellB numbers versus their asymptotics:

Compute the first 10 complementary Bell numbers:

Compare with an expression in terms of the Stirling number of the second kind:

Verify an expression for the Bell number in terms of a Hessenberg determinant for the first few cases:

The Bell numbers BellB[n] can be characterized as the unique set of numbers such that two certain Hankel determinants made from these numbers are both equal to BarnesG[n+2]. Verify for the first few cases:

## Properties & Relations(7)

The exponential generating function for BellB:

Compare with the explicit summation formula:

Sum can give results involving BellB:

The moment of a PoissonDistribution is given by the Bell 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 with the result of BellB:

Compute Bell numbers using generalized Bell polynomials:

Compute Bell polynomials using generalized Bell polynomials:

FindSequenceFunction can recognize the BellB sequence:

## Possible Issues(1)

The first argument of BellB must be a non-negative integer:

## Neat Examples(1)

Integral representation for Bell numbers by Cesàro:

Wolfram Research (2007), BellB, Wolfram Language function, https://reference.wolfram.com/language/ref/BellB.html.

#### 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

#### BibTeX

@misc{reference.wolfram_2024_bellb, author="Wolfram Research", title="{BellB}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/BellB.html}", note=[Accessed: 17-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_bellb, organization={Wolfram Research}, title={BellB}, year={2007}, url={https://reference.wolfram.com/language/ref/BellB.html}, note=[Accessed: 17-July-2024 ]}