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 $\left(-1+\mathrm{xD}\right)\left(x\right)\left(x^2\mathrm{D}+4\right)\left(\mathrm{D}\right)$.

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 $\frac{2}{x}+x\mathrm{D}+\left(x+1\right){\mathrm{D}}^2+{\mathrm{D}}^3$.

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.

$\mathrm{expr}≔\mathrm{\Psi }\left(n\right)+\mathrm{\Psi }\left(n+1\right)+n$

$\mathrm{expr}≔\mathrm{\Psi }\left(n\right)+\mathrm{\Psi }\left(n+1\right)+n$

$L≔\mathrm{LinearOperators}\left[\mathrm{FactoredAnnihilator}\right]\left(\mathrm{expr}\,n\,'\mathrm{shift}'\right)$

$L≔\mathrm{FactoredOrePoly}\left(\left[-\frac{n+2}{n+4}\,1\right]\,\left[-\frac{n^2+2n+1}{\left(n+3\right)\left(n+2\right)}\,1\right]\,\left[-\frac{n}{n+1}\,1\right]\,\left[\mathrm{-1}\,1\right]\right)$

$\mathrm{LM}≔\mathrm{LinearOperators}\left[\mathrm{MinimalAnnihilator}\right]\left(L\,\mathrm{expr}\,n\,'\mathrm{shift}'\right)$

$\mathrm{LM}≔\mathrm{OrePoly}\left(\frac{n\left(n^2+5n+5\right)}{n^3+5n^2+7n+2}\,-\frac{2\left(n^3+5n^2+6n+1\right)}{n^3+5n^2+7n+2}\,1\right)$

$\mathrm{normal}\left(\mathrm{LinearOperators}\left[\mathrm{Apply}\right]\left(\mathrm{LM}\,\mathrm{expr}\,n\,'\mathrm{shift}'\right)\,'\mathrm{expanded}'\right)$

$0$

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.