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LinearOperators

  

MinimalAnnihilator

  

construct the minimal annihilator

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

MinimalAnnihilator(L, expr, x, case)

Parameters

L

-

completely factored Ore operator

expr

-

Maple expression

x

-

name of the independent variable

case

-

parameter indicating the case of the equation ('differential' or 'shift')

Description

• 

Given a factored Ore operator L that is an annihilator for the expression expr, the LinearOperators[MinimalAnnihilator] function returns the minimal annihilator in non-factored form for expr.

• 

A completely factored Ore operator is an operator that can be factored into a product of factors of degree at most one.

• 

A completely factored Ore operator is represented by a structure that consists of the keyword FactoredOrePoly and a sequence of lists. Each list consists of two elements and describes a first degree factor. The first element provides the zero degree coefficient and the second element provides the first degree coefficient. For example, in the differential case with a differential operator D, FactoredOrePoly([-1, x], [x, 0], [4, x^2], [0, 1]) describes the operator 1+xDxDx2+4D.

• 

An Ore operator is a structure that consists of the keyword OrePoly with a sequence of coefficients starting with the one of degree zero. The coefficients must be rational functions in x. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator 2x+xD+x+1D2+D3.

• 

There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].

• 

The expression expr must be a d'Alembertian term. The main property of a d'Alembertian term is that it is annihilated by a linear operator that can be written as a composition of operators of the first degree. The set of d'Alembertian terms has a ring structure. The package recognizes some basic d'Alembertian terms and their ring-operation closure terms. The result of the substitution of a rational term for the independent variable in the d'Alembertian term is also a d'Alembertian term.

  

Note: The operator L must annihilate expr, that is, satisfy L(expr)=0.

Examples

exprΨn+Ψn+1+n

expr:=Ψn+Ψn+1+n

(1)

LLinearOperators[FactoredAnnihilator]expr,n,'shift'

L:=FactoredOrePolyn+2n+4,1,n2+2n+1n+3n+2,1,nn+1,1,1,1

(2)

LMLinearOperators[MinimalAnnihilator]L,expr,n,'shift'

LM:=OrePolynn2+5n+5n3+5n2+7n+2,2n3+5n2+6n+1n3+5n2+7n+2,1

(3)

normalLinearOperators[Apply]LM,expr,n,'shift','expanded'

0

(4)

References

  

Abramov, S. A., and Zima, E. V. "Minimal Completely Factorable Annihilators." In Proceedings of ISSAC '97, pp. 290-297. Edited by Wolfgang Kuchlin. New York: ACM Press, 1997.

See Also

LinearOperators

LinearOperators[Apply]

LinearOperators[converters]

LinearOperators[FactoredAnnihilator]

 


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