MinimalAnnihilator - Maple Help

LinearOperators

 MinimalAnnihilator
 construct the minimal annihilator

 Calling Sequence MinimalAnnihilator(L, expr, x, case)

Parameters

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

Description

 • 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.

Examples

 > $\mathrm{expr}≔\mathrm{\Psi }\left(n\right)+\mathrm{\Psi }\left(n+1\right)+n$
 ${\mathrm{expr}}{≔}{\mathrm{\Psi }}{}\left({n}\right){+}{\mathrm{\Psi }}{}\left({n}{+}{1}\right){+}{n}$ (1)
 > $L≔\mathrm{LinearOperators}\left[\mathrm{FactoredAnnihilator}\right]\left(\mathrm{expr},n,'\mathrm{shift}'\right)$
 ${L}{≔}{\mathrm{FactoredOrePoly}}{}\left(\left[{-}\frac{{n}{+}{2}}{{n}{+}{4}}{,}{1}\right]{,}\left[{-}\frac{{{n}}^{{2}}{+}{2}{}{n}{+}{1}}{\left({n}{+}{3}\right){}\left({n}{+}{2}\right)}{,}{1}\right]{,}\left[{-}\frac{{n}}{{n}{+}{1}}{,}{1}\right]{,}\left[{-1}{,}{1}\right]\right)$ (2)
 > $\mathrm{LM}≔\mathrm{LinearOperators}\left[\mathrm{MinimalAnnihilator}\right]\left(L,\mathrm{expr},n,'\mathrm{shift}'\right)$
 ${\mathrm{LM}}{≔}{\mathrm{OrePoly}}{}\left(\frac{{n}{}\left({{n}}^{{2}}{+}{5}{}{n}{+}{5}\right)}{{{n}}^{{3}}{+}{5}{}{{n}}^{{2}}{+}{7}{}{n}{+}{2}}{,}{-}\frac{{2}{}\left({{n}}^{{3}}{+}{5}{}{{n}}^{{2}}{+}{6}{}{n}{+}{1}\right)}{{{n}}^{{3}}{+}{5}{}{{n}}^{{2}}{+}{7}{}{n}{+}{2}}{,}{1}\right)$ (3)
 > $\mathrm{normal}\left(\mathrm{LinearOperators}\left[\mathrm{Apply}\right]\left(\mathrm{LM},\mathrm{expr},n,'\mathrm{shift}'\right),'\mathrm{expanded}'\right)$
 ${0}$ (4)

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.