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LinearOperators

 FactoredAnnihilator
 construct completely factorable annihilator

 Calling Sequence FactoredAnnihilator(expr, x, case)

Parameters

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

Description

 • Given an algebraic expression expr, the LinearOperators[FactoredAnnihilator] function returns a completely factored Ore operator that is an annihilator for expr. That is, applying this operator to expr yields zero. If such an operator does not exist, the function returns $-1$.
 • 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 $\left(-1+\mathrm{xD}\right)\left(x\right)\left(\mathrm{D}{x}^{2}+4\right)\left(\mathrm{D}\right)$.
 • There are routines in the package that convert between Ore operators and the corresponding Maple expressions. See LinearOperators[converters].
 • To get the completely factorable annihilator, 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

 > $\mathrm{expr}≔{ⅇ}^{\frac{x}{x-1}}+\sqrt{{x}^{2}+1}$
 ${\mathrm{expr}}{≔}{{ⅇ}}^{\frac{{x}}{{x}{-}{1}}}{+}\sqrt{{{x}}^{{2}}{+}{1}}$ (1)
 > $L≔\mathrm{LinearOperators}[\mathrm{FactoredAnnihilator}]\left(\mathrm{expr},x,'\mathrm{differential}'\right)$
 ${L}{≔}{\mathrm{FactoredOrePoly}}{}\left(\left[\frac{{{x}}^{{4}}{+}{6}{}{{x}}^{{2}}{-}{2}{}{x}{+}{3}}{\left({{x}}^{{3}}{-}{{x}}^{{2}}{+}{x}{+}{1}\right){}\left({{x}}^{{2}}{+}{1}\right){}\left({x}{-}{1}\right)}{,}{1}\right]{,}\left[\frac{{1}}{{\left({x}{-}{1}\right)}^{{2}}}{,}{1}\right]\right)$ (2)
 > $\mathrm{LinearOperators}[\mathrm{Apply}]\left(L,\mathrm{expr},x,'\mathrm{differential}'\right)$
 ${0}$ (3)

References

 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.