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Calling Sequence
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BellB()
IncompleteBellB()
IncompleteBellB[DiamondConvolution]()
CompleteBellB()
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Parameters
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non-negative integers, or algebraic expressions representing them
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the main variables of the polynomials, or algebraic expressions representing them
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Description
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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.
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For the definition of the IncompleteBellB polynomials, consider a sequence with , with which we construct the sequence
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where the element is here defined as
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Taking , the IncompleteBellB polynomials are defined in terms of an operation involving factors as
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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.
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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).
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The CompleteBellB polynomials are in turn defined in terms of the IncompleteBellB polynomials as
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When the sequence passed to CompleteBellB contains less than elements, the missing ones will be assumed equal to zero.
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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.
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The Bell polynomials appear in various applications, including for instance Faà di Bruno's formula
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where represents the derivative of evaluated at ; the exponential of a formal power series
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and in the following exponential generating function
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Examples
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The Bell functions only evaluate to a polynomial when the arguments specifying the degree are positive integers
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A sequence with the values of for
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The IncompleteBellB polynomials have a special form for some particular values of the function's parameters. For illustration purposes consider the generic sequence
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For and , or and , or , IncompleteBellB is equal to 0
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For , the following identity holds
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If for all , the following identity holds
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If for all , the following identity, here expressed in terms of the inert sequence %seq, holds
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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
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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
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These are the first 5 elements of , a diamond operation involving 3 factors and the value of
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The value of is obtained by adding the values of for as explained in the Description
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References
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Bell, E. T. "Exponential Polynomials", Ann. Math., Vol. 35 (1934): 258-277.
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Compatibility
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The BellB, IncompleteBellB and CompleteBellB commands were introduced in Maple 15.
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