- 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, .
Examplesopen allclose all
BellB numbers versus their asymptotics:
Compute the first 10 complementary Bell numbers:
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 (8)
Use IntegerPartitions to directly sum over terms that satisfy the constraints on indices:
The exponential generating function for BellB:
Possible Issues (1)
The first argument of BellB must be a non-negative integer:
Wolfram Research (2007), BellB, Wolfram Language function, https://reference.wolfram.com/language/ref/BellB.html.
Wolfram Language. 2007. "BellB." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BellB.html.
Wolfram Language. (2007). BellB. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BellB.html