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BellB

the Bell polynomials

IncompleteBellB

the incomplete Bell polynomials

CompleteBellB

the complete Bell polynomials

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

BellB()

IncompleteBellB()

IncompleteBellB[DiamondConvolution]()

CompleteBellB()

Parameters

-

non-negative integers, or algebraic expressions representing them

-

the main variables of the polynomials, or algebraic expressions representing them

Description

• 

The BellB, IncompleteBellB, and CompleteBellB respectively represent the Bell polynomials, the incomplete Bell polynomials - also called Bell polynomials of the second kind - and the complete Bell polynomials. For the Bell numbers, see bell.

• 

The BellB polynomials are polynomials of degree  defined in terms of the Stirling numbers of the second kind as

• 

For the definition of the IncompleteBellB polynomials, consider a sequence  with , with which we construct the sequence

  

where the  element is here defined as

  

Taking , the IncompleteBellB polynomials are defined in terms of an operation  involving  factors as

  

The output of IncompleteBellB is thus a multivariable polynomial of degree  in the  variables. Note that the right-hand side of this formula involves only the first  elements of the sequence ; so in the left-hand side only the first   are relevant, and all those not given in the input to IncompleteBellB will be assumed equal to zero.

• 

To compute the first  elements of the sequence obtained by performing this diamond operation  between  factors you can use the IncompleteBellB:-DiamondConvolution command. This command makes use of the first  elements of the sequence  and returns a sequence of  elements, where the first  are equal to zero and the remaining  are all polynomials of degree  in the  variables. Note that, unlike IncompleteBellB, IncompleteBellB:-DiamondConvolution expects the sequence  enclosed as a list as third argument (see the Examples section).

• 

The CompleteBellB polynomials are in turn defined in terms of the IncompleteBellB polynomials as

  

When the sequence  passed to CompleteBellB contains less than  elements, the missing ones will be assumed equal to zero.

• 

All of CompleteBellB, IncompleteBellB and IncompleteBellB:-DiamondConvolution accept inert sequences constructed with %seq or the quoted 'seq' functions as part of the  arguments, in which case they return unevaluated, echoing the input.

• 

The Bell polynomials appear in various applications, including for instance Faà di Bruno's formula

  

where  represents the  derivative of  evaluated at ; the exponential of a formal power series

  

and in the following exponential generating function

Examples

The Bell functions only evaluate to a polynomial when the arguments specifying the degree are positive integers

(1)

(2)

(3)

A sequence with the values of  for

(4)

The IncompleteBellB polynomials have a special form for some particular values of the function's parameters. For illustration purposes consider the generic sequence

(5)

(6)

For  and , or  and , or , IncompleteBellB is equal to 0

(7)

(8)

For , the following identity holds  

(9)

(10)

If  for all , the following identity holds

(11)

(12)

If  for all , the following identity, here expressed in terms of the inert sequence %seq, holds

(13)

(14)

(15)

The diamond operation that enters the definition of IncompleteBellB can be invoked directly as IncompleteBellB:-DiamondConvolution. These are the first 4 elements of , a diamond operation involving 2 factors

(16)

Note that when calling IncompleteBellB:-DiamondConvolution, you pass the sequence  enclosed in a list. The value of  is equal to the 4th element of the above sequence divided by

(17)

These are the first 5 elements of , a diamond operation involving 3 factors and the value of

(18)

(19)

The value of  is obtained by adding the values of  for  as explained in the Description

(20)

References

  

Bell, E. T. "Exponential Polynomials", Ann. Math., Vol. 35 (1934): 258-277.

Compatibility

• 

The BellB, IncompleteBellB and CompleteBellB commands were introduced in Maple 15.

• 

For more information on Maple 15 changes, see Updates in Maple 15.

See Also

bell

FunctionAdvisor

Stirling2

 


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