construct the minimal annihilator
MinimalAnnihilator(L, expr, x, case)
completely factored Ore operator
name of the independent variable
parameter indicating the case of the equation ('differential' or 'shift')
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+xD⁡x⁡x2⁢D+4⁡D.
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+x⁢D+x+1⁢D2+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.
expr ≔ Ψ⁡n+Ψ⁡n+1+n
L ≔ LinearOperatorsFactoredAnnihilator⁡expr,n,'shift'
LM ≔ LinearOperatorsMinimalAnnihilator⁡L,expr,n,'shift'
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.
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