the Bell polynomials
the incomplete Bell polynomials
the complete Bell polynomials
non-negative integers, or algebraic expressions representing them
the main variables of the polynomials, or algebraic expressions representing them
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
For the definition of the IncompleteBellB polynomials, consider a sequence zn with n=1,2,3,..., with which we construct the sequence
where the nth element is here defined as
Taking z⋄z⋄z=z⋄z⋄z, the IncompleteBellB polynomials are defined in terms of an operation z⋄...⋄z involving k factors as
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 n−k+1 elements of the sequence zj; so in the left-hand side only the first n−k+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 n−k+1 elements of the sequence zj and returns a sequence of n elements, where the first k−1 are equal to zero and the remaining n−k+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
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
where fk⁢g⁡x represents the kth derivative of f⁡x evaluated at g⁡x; the exponential of a formal power series
and in the following exponential generating function
The Bell functions only evaluate to a polynomial when the arguments specifying the degree are positive integers
A sequence with the values of BellB⁡n,z for n=0..3
The IncompleteBellB polynomials have a special form for some particular values of the function's parameters. For illustration purposes consider the generic sequence
Z ≔ z1,z2,z3,z4,z5
For n=0 and 0<k, or 0<n and k=0, or n<k, IncompleteBellB is equal to 0
For n=k, the following identity holds
If zj=1 for all j, the following identity holds
If zj=j! for all j=1,..,n−k+1, the following identity, here expressed in terms of the inert sequence %seq, holds
The diamond operation that enters the definition of IncompleteBellB can be invoked directly as IncompleteBellB:-DiamondConvolution. These are the first 4 elements of z⋄z⁢, a diamond operation involving 2 factors
Note that when calling IncompleteBellB:-DiamondConvolution, you pass the sequence Z enclosed in a list. The value of IncompleteBellB⁡4,2,Z is equal to the 4th element of the above sequence divided by 2!
These are the first 5 elements of z⋄z⋄z, a diamond operation involving 3 factors and the value of IncompleteBellB⁡5,3,Z
The value of CompleteBellB⁡5,Z is obtained by adding the values of IncompleteBellB⁡5,k,Z for k=1..5 as explained in the Description
Bell, E. T. "Exponential Polynomials", Ann. Math., Vol. 35 (1934): 258-277.
The BellB, IncompleteBellB and CompleteBellB commands were introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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