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OreTools

 Apply
 apply an Ore polynomial to an expression

 Calling Sequence Apply(Poly, expr, A)

Parameters

 Poly - Ore polynomial; to define an Ore polynomial, use the OrePoly structure. expr - Maple expression A - Ore algebra; to define an Ore algebra, use the SetOreRing function.

Description

 • The Apply(Poly, expr, A) calling sequence applies the Ore polynomial Poly to the expression expr.

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)
 > $L≔\mathrm{OrePoly}\left({x}^{2}-1,x,{x}^{2}\right)$
 ${L}{:=}{\mathrm{OrePoly}}{}\left({{x}}^{{2}}{-}{1}{,}{x}{,}{{x}}^{{2}}\right)$ (2)
 > $\mathrm{Apply}\left(L,f\left(x\right),A\right)$
 $\left({{x}}^{{2}}{-}{1}\right){}{f}{}\left({x}\right){+}{x}{}\left(\frac{{ⅆ}}{{ⅆ}{x}}{}{f}{}\left({x}\right)\right){+}{{x}}^{{2}}{}\left(\frac{{{ⅆ}}^{{2}}}{{ⅆ}{{x}}^{{2}}}{}{f}{}\left({x}\right)\right)$ (3)

Define the shift algebra.

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