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LinearOperators

 DEToOrePoly
 convert a linear ordinary differential equation to an OrePoly structure
 REToOrePoly
 convert a linear recurrence equation to an OrePoly structure
 OrePolyToDE
 convert an OrePoly structure to a linear ordinary differential equation
 OrePolyToRE
 convert an OrePoly structure to a linear recurrence equation
 FactoredOrePolyToDE
 convert a FactoredOrePoly structure to a linear ordinary differential equation
 FactoredOrePolyToRE
 convert a FactoredOrePoly structure to a linear recurrence equation
 FactoredOrePolyToOrePoly
 convert a FactoredOrePoly structure to a OrePoly structure Calling Sequence DEToOrePoly(eq,f) REToOrePoly(eq,f) OrePolyToDE(L,f) OrePolyToRE(L,f) FactoredOrePolyToDE(M,f) FactoredOrePolyToRE(M,f) FactoredOrePolyToOrePoly(M,var,case) Parameters

 eq - left hand side of a linear equation (either differential or recurrence) f - function from eq, for example, f(x) L - Ore operator M - factored Ore operator var - name of the independent variable case - parameter indicating the case of the equation ('differential' or 'shift') Description

 • 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 $\frac{2}{x}+x\mathrm{D}+\left(x+1\right){\mathrm{D}}^{2}+{\mathrm{D}}^{3}$. Examples

 > $\mathrm{poly}≔\mathrm{FactoredOrePoly}\left(\left[{x}^{5},1+x\right],\left[x,-1\right]\right)$
 ${\mathrm{poly}}{≔}{\mathrm{FactoredOrePoly}}{}\left(\left[{{x}}^{{5}}{,}{x}{+}{1}\right]{,}\left[{x}{,}{-1}\right]\right)$ (1)
 > $\mathrm{ode}≔\mathrm{LinearOperators}\left[\mathrm{FactoredOrePolyToDE}\right]\left(\mathrm{poly},y\left(x\right)\right)$
 ${\mathrm{ode}}{≔}{y}{}\left({x}\right){}{{x}}^{{6}}{-}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{{x}}^{{5}}{+}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{{x}}^{{2}}{+}{y}{}\left({x}\right){}{x}{+}\left(\frac{{ⅆ}}{{ⅆ}{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{-}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)\right){}{x}{+}{y}{}\left({x}\right){-}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{y}{}\left({x}\right)$ (2)
 > $\mathrm{LinearOperators}\left[\mathrm{DEToOrePoly}\right]\left(\mathrm{ode},y\left(x\right)\right)$
 ${\mathrm{OrePoly}}{}\left({{x}}^{{6}}{+}{x}{+}{1}{,}{-}{{x}}^{{5}}{+}{{x}}^{{2}}{+}{x}{,}{-}{x}{-}{1}\right)$ (3)
 > $\mathrm{lre}≔\mathrm{LinearOperators}\left[\mathrm{FactoredOrePolyToRE}\right]\left(\mathrm{poly},y\left(x\right)\right)$
 ${\mathrm{lre}}{≔}{y}{}\left({x}\right){}{{x}}^{{6}}{-}{y}{}\left({x}{+}{1}\right){}{{x}}^{{5}}{+}{y}{}\left({x}{+}{1}\right){}{{x}}^{{2}}{+}{2}{}{x}{}{y}{}\left({x}{+}{1}\right){-}{x}{}{y}{}\left({x}{+}{2}\right){+}{y}{}\left({x}{+}{1}\right){-}{y}{}\left({x}{+}{2}\right)$ (4)
 > $\mathrm{LinearOperators}\left[\mathrm{REToOrePoly}\right]\left(\mathrm{lre},y\left(x\right)\right)$
 ${\mathrm{OrePoly}}{}\left({{x}}^{{6}}{,}{-}{{x}}^{{5}}{+}{{x}}^{{2}}{+}{2}{}{x}{+}{1}{,}{-}{x}{-}{1}\right)$ (5)
 > $\mathrm{lre}≔\mathrm{LinearOperators}\left[\mathrm{OrePolyToRE}\right]\left(\mathrm{OrePoly}\left(x,{x}^{3}+x-2,{x}^{5}\right),y\left(x\right)\right)$
 ${\mathrm{lre}}{≔}{{x}}^{{5}}{}{y}{}\left({x}{+}{2}\right){+}{y}{}\left({x}{+}{1}\right){}{{x}}^{{3}}{+}{x}{}{y}{}\left({x}{+}{1}\right){+}{y}{}\left({x}\right){}{x}{-}{2}{}{y}{}\left({x}{+}{1}\right)$ (6)
 > $\mathrm{LinearOperators}\left[\mathrm{REToOrePoly}\right]\left(\mathrm{lre},y\left(x\right)\right)$
 ${\mathrm{OrePoly}}{}\left({x}{,}{{x}}^{{3}}{+}{x}{-}{2}{,}{{x}}^{{5}}\right)$ (7)
 > $L≔\mathrm{FactoredOrePoly}\left(\left[\frac{3}{2x},1\right],\left[\frac{1}{2x},1\right],\left[-\frac{1}{2x},1\right]\right)$
 ${L}{≔}{\mathrm{FactoredOrePoly}}{}\left(\left[\frac{{3}}{{2}{}{x}}{,}{1}\right]{,}\left[\frac{{1}}{{2}{}{x}}{,}{1}\right]{,}\left[{-}\frac{{1}}{{2}{}{x}}{,}{1}\right]\right)$ (8)
 > $\mathrm{LinearOperators}\left[\mathrm{FactoredOrePolyToOrePoly}\right]\left(L,x,'\mathrm{differential}'\right)$
 ${\mathrm{OrePoly}}{}\left({-}\frac{{1}}{{8}{}{{x}}^{{3}}}{,}\frac{{1}}{{4}{}{{x}}^{{2}}}{,}\frac{{3}}{{2}{}{x}}{,}{1}\right)$ (9)