The LinearOperators[Apply] function applies the polynomial L to the expression expr. The polynomial can be an OrePoly or a FactoredOrePoly structure.

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].

This command is part of the LinearOperators package, so it can be used in the form Apply(..) only after executing the command with(LinearOperators).  However, it can always be used in the form LinearOperators[Apply](..).

$\mathrm{expr}≔\exp \left(\frac{x}{x-1}\right)+\mathrm{sqrt}\left(x^2+1\right)$

$\mathrm{expr}≔{\ⅇ}^{\frac{x}{x-1}}+\sqrt{x^2+1}$

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

$L≔\mathrm{FactoredOrePoly}\left(\left[\frac{x^4+6x^2-2x+3}{\left(x^3-x^2+x+1\right)\left(x^2+1\right)\left(x-1\right)}\,1\right]\,\left[\frac{1}{{\left(x-1\right)}^2}\,1\right]\right)$

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

$0$

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.