Given a completely factored Ore operator U and a non-factored Ore operator V, the LinearOperators[FactoredGCRD] function returns the greatest common right divisor (GCRD) in the completely factored form.

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].

$a≔\mathrm{FactoredOrePoly}\left(\left[-1\,x\right]\,\left[3\,x\right]\right)$

$a≔\mathrm{FactoredOrePoly}\left(\left[\mathrm{-1}\,x\right]\,\left[3\,x\right]\right)$

$b≔\mathrm{OrePoly}\left(0\,0\,2x^3+4x^2\,x^4\right)$

$b≔\mathrm{OrePoly}\left(0\,0\,2x^3+4x^2\,x^4\right)$

$\mathrm{LinearOperators}\left[\mathrm{FactoredGCRD}\right]\left(a\,b\,x\,'\mathrm{differential}'\right)$

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

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