The OrePoly Structure - Maple Programming Help

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The OrePoly Structure

 • An Ore polynomial is represented by an OrePoly structure. It consists of the constructor OrePoly with a sequence of coefficients starting with the one of degree zero. For example, in the differential case with the differential operator D, OrePoly(2/x, x, x+1, 1) represents the operator 2/x+xD+(x+1)D^2+D^3.
 • For a brief review of pseudo-linear algebra (also known as Ore algebra), see OreAlgebra.

Examples

 > $\mathrm{with}\left(\mathrm{OreTools}\right):$

Define the differential algebra.

 > $A≔\mathrm{SetOreRing}\left(x,'\mathrm{differential}'\right)$
 ${A}{:=}{\mathrm{UnivariateOreRing}}{}\left({x}{,}{\mathrm{differential}}\right)$ (1)
 > $\mathrm{Poly}≔\mathrm{OrePoly}\left(\frac{2\left(3x+1\right)}{{x}^{2}\left(4+27x\right)},-\frac{2}{x\left(4+27x\right)},1\right)$
 ${\mathrm{Poly}}{:=}{\mathrm{OrePoly}}{}\left(\frac{{2}{}\left({3}{}{x}{+}{1}\right)}{{{x}}^{{2}}{}\left({4}{+}{27}{}{x}\right)}{,}{-}\frac{{2}}{{x}{}\left({4}{+}{27}{}{x}\right)}{,}{1}\right)$ (2)
 > $\mathrm{Apply}\left(\mathrm{Poly},f\left(x\right),A\right)$
 $\frac{{2}{}\left({3}{}{x}{+}{1}\right){}{f}{}\left({x}\right)}{{{x}}^{{2}}{}\left({4}{+}{27}{}{x}\right)}{-}\frac{{2}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)\right)}{{x}{}\left({4}{+}{27}{}{x}\right)}{+}\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)$ (3)

Define the shift algebra.

 > $A≔\mathrm{SetOreRing}\left(n,'\mathrm{shift}'\right)$
 ${A}{:=}{\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{shift}}\right)$ (4)
 > $\mathrm{Poly}≔\mathrm{OrePoly}\left(1,-2,-2,1\right)$
 ${\mathrm{Poly}}{:=}{\mathrm{OrePoly}}{}\left({1}{,}{-}{2}{,}{-}{2}{,}{1}\right)$ (5)
 > $\mathrm{Apply}\left(\mathrm{Poly},s\left(n\right),A\right)$
 ${s}{}\left({n}\right){-}{2}{}{s}{}\left({n}{+}{1}\right){-}{2}{}{s}{}\left({n}{+}{2}\right){+}{s}{}\left({n}{+}{3}\right)$ (6)

Define the q-shift algebra.

 > $A≔\mathrm{SetOreRing}\left(\left[x,q\right],'\mathrm{qshift}'\right)$
 ${A}{:=}{\mathrm{UnivariateOreRing}}{}\left({x}{,}{\mathrm{qshift}}\right)$ (7)
 > $\mathrm{Poly}≔\mathrm{OrePoly}\left(-q\left(1-qx\right),1\right)$
 ${\mathrm{Poly}}{:=}{\mathrm{OrePoly}}{}\left({-}{q}{}\left({-}{q}{}{x}{+}{1}\right){,}{1}\right)$ (8)
 > $\mathrm{Apply}\left(\mathrm{Poly},s\left(x\right),A\right)$
 ${-}{q}{}\left({-}{q}{}{x}{+}{1}\right){}{s}{}\left({x}\right){+}{s}{}\left({q}{}{x}\right)$ (9)