BellB

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`BellB`

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BellB[n]

gives the Bell number .

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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 $e^((e^t-1)x)=sum_(n=0)^(infty)(TemplateBox[{n, x}, BellB2]t^n)/(n!)$ .
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 $B_(n+1)=sum_(k=0)^nTemplateBox[{n, k}, Binomial]B_k$ . 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 $TemplateBox[{n, x}, BellB2]=sum_(k=0)^nY_(n,k)(x,...,x)$ and . Letting denote the Stirling number of the second kind, returned by StirlingS2, $B_n=B_n(1)=sum_(k=0)^nTemplateBox[{n, k}, StirlingS2]$ .

Examples

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Basic Examples (2)Summary of the most common use cases

The tenth Bell number:

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https://wolfram.com/xid/0wgvxr5-mhpvry

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The fifth Bell polynomial:

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https://wolfram.com/xid/0wgvxr5-cifqef

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Scope (5)Survey of the scope of standard use cases

Evaluate numerically:

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https://wolfram.com/xid/0wgvxr5-onyluq

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

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https://wolfram.com/xid/0wgvxr5-bn0shu

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BellB threads element-wise over lists:

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https://wolfram.com/xid/0wgvxr5-cl7plc

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BellB can be applied to a power series:

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https://wolfram.com/xid/0wgvxr5-iemucp

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TraditionalForm formatting:

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https://wolfram.com/xid/0wgvxr5-7pz8n

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https://wolfram.com/xid/0wgvxr5-g690em

Applications (4)Sample problems that can be solved with this function

BellB numbers versus their asymptotics:

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https://wolfram.com/xid/0wgvxr5-fnjyyy

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Compute the first 10 complementary Bell numbers:

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https://wolfram.com/xid/0wgvxr5-hzd9e

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

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https://wolfram.com/xid/0wgvxr5-n8dndn

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

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https://wolfram.com/xid/0wgvxr5-mu795p

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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: