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

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.

$\mathrm{expr}≔x\ln \left(x\right)-x+1$

$\mathrm{expr}≔x\ln \left(x\right)-x+1$

$L≔\mathrm{LinearOperators}\left[\mathrm{FactoredMinimalAnnihilator}\right]\left(\mathrm{expr}\,x\,'\mathrm{differential}'\right)$

$L≔\mathrm{FactoredOrePoly}\left(\left[-\frac{1}{\left(x-1\right)x}\,1\right]\,\left[-\frac{1}{x}\,1\right]\right)$

$\mathrm{LinearOperators}\left[\mathrm{Apply}\right]\left(L\,\mathrm{expr}\,x\,'\mathrm{differential}'\right)$

$0$

$\mathrm{expr}≔\mathrm{\Gamma }\left(n\right)n^2$

$\mathrm{expr}≔\mathrm{\Gamma }\left(n\right)n^2$

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

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

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

$0$

Abramov, S.A., and  Zima, E.V. "Minimal Completely Factorable Annihilators." Proc. ISSAC'97. 1997.