The LinearOperators[DEToOrePoly] and LinearOperators[REToOrePoly] functions return an Ore operator K such that eq = K(f). The LinearOperators[OrePolyToDE], LinearOperators[OrePolyToRE], LinearOperators[FactoredOrePolyToDE], and LinearOperators[FactoredOrePolyToRE] functions apply the operator (L or M) to the function f. The LinearOperators[FactoredOrePolyToOrePoly] function converts an Ore polynomial in factored form, that is, a FactoredOrePoly structure, to an Ore polynomial in expanded form, that is, an OrePoly 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 $\frac{2}{x}+x\mathrm{D}+\left(x+1\right){\mathrm{D}}^2+{\mathrm{D}}^3$.

$\mathrm{poly}≔\mathrm{FactoredOrePoly}\left(\left[x^5\,1+x\right]\,\left[x\,-1\right]\right)$

$\mathrm{poly}≔\mathrm{FactoredOrePoly}\left(\left[x^5\,x+1\right]\,\left[x\,\mathrm{-1}\right]\right)$

$\mathrm{ode}≔\mathrm{LinearOperators}\left[\mathrm{FactoredOrePolyToDE}\right]\left(\mathrm{poly}\,y\left(x\right)\right)$

$\mathrm{ode}≔y\left(x\right)x^6-\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x^5+\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x^2+y\left(x\right)x+\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)x-\left(\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)\right)x+y\left(x\right)-\frac{{\ⅆ}^2}{\ⅆx^2}y\left(x\right)$

$\mathrm{LinearOperators}\left[\mathrm{DEToOrePoly}\right]\left(\mathrm{ode}\,y\left(x\right)\right)$

$\mathrm{OrePoly}\left(x^6+x+1\,-x^5+x^2+x\,-x-1\right)$

$\mathrm{lre}≔\mathrm{LinearOperators}\left[\mathrm{FactoredOrePolyToRE}\right]\left(\mathrm{poly}\,y\left(x\right)\right)$

$\mathrm{lre}≔y\left(x\right)x^6-y\left(x+1\right)x^5+y\left(x+1\right)x^2+2xy\left(x+1\right)-xy\left(x+2\right)+y\left(x+1\right)-y\left(x+2\right)$

$\mathrm{LinearOperators}\left[\mathrm{REToOrePoly}\right]\left(\mathrm{lre}\,y\left(x\right)\right)$

$\mathrm{OrePoly}\left(x^6\,-x^5+x^2+2x+1\,-x-1\right)$

$\mathrm{lre}≔\mathrm{LinearOperators}\left[\mathrm{OrePolyToRE}\right]\left(\mathrm{OrePoly}\left(x\,x^3+x-2\,x^5\right)\,y\left(x\right)\right)$

$\mathrm{lre}≔x^{5}y\left(x+2\right)+y\left(x+1\right)x^3+y\left(x\right)x+xy\left(x+1\right)-2y\left(x+1\right)$

$\mathrm{LinearOperators}\left[\mathrm{REToOrePoly}\right]\left(\mathrm{lre}\,y\left(x\right)\right)$

$\mathrm{OrePoly}\left(x\,x^3+x-2\,x^5\right)$

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

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

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

$\mathrm{OrePoly}\left(-\frac{1}{8x^3}\,\frac{1}{4x^2}\,\frac{3}{2x}\,1\right)$