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LinearOperators

 FactoredMinimalAnnihilator
 construct the minimal annihilator in the completely factored form

 Calling Sequence FactoredMinimalAnnihilator(expr, x, case)

Parameters

 expr - Maple expression x - name of the independent variable case - parameter indicating the case of the equation ('differential' or 'shift')

Description

 • 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.

Examples

 > $\mathrm{expr}≔x\mathrm{ln}\left(x\right)-x+1$
 ${\mathrm{expr}}{≔}{x}{}{\mathrm{ln}}{}\left({x}\right){-}{x}{+}{1}$ (1)
 > $L≔\mathrm{LinearOperators}\left[\mathrm{FactoredMinimalAnnihilator}\right]\left(\mathrm{expr},x,'\mathrm{differential}'\right)$
 ${L}{≔}{\mathrm{FactoredOrePoly}}{}\left(\left[{-}\frac{{1}}{\left({x}{-}{1}\right){}{x}}{,}{1}\right]{,}\left[{-}\frac{{1}}{{x}}{,}{1}\right]\right)$ (2)
 > $\mathrm{LinearOperators}\left[\mathrm{Apply}\right]\left(L,\mathrm{expr},x,'\mathrm{differential}'\right)$
 ${0}$ (3)
 > $\mathrm{expr}≔\mathrm{\Gamma }\left(n\right){n}^{2}$
 ${\mathrm{expr}}{≔}{\mathrm{\Gamma }}{}\left({n}\right){}{{n}}^{{2}}$ (4)
 > $L≔\mathrm{LinearOperators}\left[\mathrm{FactoredMinimalAnnihilator}\right]\left(\mathrm{expr},n,'\mathrm{shift}'\right)$
 ${L}{≔}{\mathrm{FactoredOrePoly}}{}\left(\left[{-}\frac{{{n}}^{{2}}{+}{2}{}{n}{+}{1}}{{n}}{,}{1}\right]\right)$ (5)
 > $\mathrm{simplify}\left(\mathrm{LinearOperators}\left[\mathrm{Apply}\right]\left(L,\mathrm{expr},n,'\mathrm{shift}'\right)\right)$
 ${0}$ (6)

References

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