construct the minimal annihilator in the completely factored form
FactoredMinimalAnnihilator(expr, x, case)
name of the independent variable
parameter indicating the case of the equation ('differential' or 'shift')
Given a d'Alembertian term expr, the LinearOperators[FactoredMinimalAnnihilator] function returns a factored Ore operator that is the minimal annihilator in the completely factored form for expr. That is, applying this operator to expr yields zero.
A completely factored Ore operator is an operator that can be factored into a product of linear factors.
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+xD⁡x⁡x2⁢D+4⁡D.
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.
expr ≔ x⁢ln⁡x−x+1
L ≔ LinearOperatorsFactoredMinimalAnnihilator⁡expr,x,'differential'
expr ≔ GAMMA⁡n⁢n2
L ≔ LinearOperatorsFactoredMinimalAnnihilator⁡expr,n,'shift'
Abramov, S.A., and Zima, E.V. "Minimal Completely Factorable Annihilators." Proc. ISSAC'97. 1997.
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