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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(n,z)

IncompleteBellB(n,k,z1,z2,...,zn=k+1)

IncompleteBellB[DiamondConvolution](n,k,z1,z2,...,znk+1)

CompleteBellB(n,z1,z2,...,zn)

Parameters

n,k

-

non-negative integers, or algebraic expressions representing them

z,z1,...,zn

-

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 n defined in terms of the Stirling numbers of the second kind as

BellBn,z=k=0nStirling2n,kzk

• 

For the definition of the IncompleteBellB polynomials, consider a sequence zn with n=1,2,3,..., with which we construct the sequence

zz=zz1,zz2,zz3,...

  

where the nth element is here defined as

zzn=j=1n1njzjznj

  

Taking zzz=zzz, the IncompleteBellB polynomials are defined in terms of an operation z...z involving k factors as

IncompleteBellBn,k,z1,z2,...=z...znk!

  

The output of IncompleteBellB is thus a multivariable polynomial of degree k in the zj variables. Note that the right-hand side of this formula involves only the first nk+1 elements of the sequence zj; so in the left-hand side only the first nk+1 zj are relevant, and all those not given in the input to IncompleteBellB will be assumed equal to zero.

• 

To compute the first n elements of the sequence obtained by performing this diamond operation z...z between k factors you can use the IncompleteBellB:-DiamondConvolution command. This command makes use of the first nk+1 elements of the sequence zj and returns a sequence of n elements, where the first k1 are equal to zero and the remaining nk+1 are all polynomials of degree k in the zj variables. Note that, unlike IncompleteBellB, IncompleteBellB:-DiamondConvolution expects the sequence zj 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

CompleteBellBn,z1,z2,...,zn=k=1nIncompleteBellBn,k,z1,z2,...,znk+1

  

When the sequence zj passed to CompleteBellB contains less than n 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 zj 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

ⅆnⅆxnfgx=k=0nfkgxIncompleteBellBn,k,gx,gx,...,gnk+1x

  

where fkgx represents the kth derivative of fx evaluated at gx; the exponential of a formal power series

ⅇn=1anznn!=n=0znCompleteBellBn,a1,...,an

  

and in the following exponential generating function

ⅇⅇt1z=n=0BellBn,ztnn!

Examples

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

BellBn,z=k=0nStirling2n,kzk

BellBn,z=k=0nStirling2n,kzk

(1)

n=4|n=4

z4+6z3+7z2+z=k=04Stirling24,kzk

(2)

value

z4+6z3+7z2+z=z4+6z3+7z2+z

(3)

A sequence with the values of BellBn,z for n=0..3

seq'BellB'n,z=BellBn,z,n=0..3

BellB0,z=1,BellB1,z=z,BellB2,z=z2+z,BellB3,z=z3+3z2+z

(4)

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

Zz1,z2,z3,z4,z5

Z:=z1,z2,z3,z4,z5

(5)

IncompleteBellB0,0,Z

1

(6)

For n&equals;0 and 0<k, or 0<n and k&equals;0, or n<k, IncompleteBellB is equal to 0

IncompleteBellB0&comma;1&comma;Z&comma;IncompleteBellB1&comma;0&comma;Z

0&comma;0

(7)

IncompleteBellB1&comma;2&comma;Z&comma;IncompleteBellB2&comma;3&comma;Z&comma;IncompleteBellB3&comma;4&comma;Z

0&comma;0&comma;0

(8)

For n&equals;k, the following identity holds  

IncompleteBellBn&comma;n&comma;Z&equals;Z1n

IncompleteBellBn&comma;n&comma;z1&comma;z2&comma;z3&comma;z4&comma;z5&equals;z1n

(9)

n&equals;3|n&equals;3

z13&equals;z13

(10)

If zj&equals;1 for all j, the following identity holds

IncompleteBellBn&comma;k&comma;1&comma;1&comma;1&comma;1&comma;1&comma;1&comma;1&equals;Stirling2n&comma;k

IncompleteBellBn&comma;k&comma;1&comma;1&comma;1&comma;1&comma;1&comma;1&comma;1&equals;Stirling2n&comma;k

(11)

n&equals;7&comma;k&equals;4|n&equals;7&comma;k&equals;4

350&equals;350

(12)

If zj&equals;j&excl; for all j&equals;1&comma;..&comma;nk&plus;1, the following identity, here expressed in terms of the inert sequence %seq, holds

IncompleteBellBn&comma;k&comma;%seqj&excl;&comma;j&equals;1..nk&plus;1&equals;binomialn&comma;kbinomialn1&comma;k1nk&excl;

IncompleteBellBn&comma;k&comma;%seqj&excl;&comma;j&equals;1..nk&plus;1&equals;binomialn&comma;kbinomialn1&comma;k1nk&excl;

(13)

n&equals;8&comma;k&equals;3|n&equals;8&comma;k&equals;3

IncompleteBellB8&comma;3&comma;%seqj&excl;&comma;j&equals;1..6&equals;141120

(14)

value

141120&equals;141120

(15)

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

IncompleteBellB:-DiamondConvolution4&comma;2&comma;Z

0&comma;2z12&comma;6z1z2&comma;8z1z3&plus;6z22

(16)

Note that when calling IncompleteBellB:-DiamondConvolution, you pass the sequence Z enclosed in a list. The value of IncompleteBellB4&comma;2&comma;Z is equal to the 4th element of the above sequence divided by 2&excl;

IncompleteBellB4&comma;2&comma;Z

4z1z3&plus;3z22

(17)

These are the first 5 elements of zzz, a diamond operation involving 3 factors and the value of IncompleteBellB5&comma;3&comma;Z

IncompleteBellB:-DiamondConvolution5&comma;3&comma;Z

0&comma;0&comma;6z13&comma;36z12z2&comma;60z12z3&plus;90z1z22

(18)

IncompleteBellB5&comma;3&comma;Z

10z12z3&plus;15z1z22

(19)

The value of CompleteBellB5&comma;Z is obtained by adding the values of IncompleteBellB5&comma;k&comma;Z for k&equals;1..5 as explained in the Description

CompleteBellB5&comma;Z

z15&plus;10z13z2&plus;10z12z3&plus;15z1z22&plus;5z1z4&plus;10z2z3&plus;z5

(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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