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

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

$A≔\mathrm{SetOreRing}\left(x\,'\mathrm{differential}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(x\,\mathrm{differential}\right)$

$L≔\mathrm{OrePoly}\left(x^2-1\,x\,x^2\right)$

$L≔\mathrm{OrePoly}\left(x^2-1\,x\,x^2\right)$

$\mathrm{Apply}\left(L\,f\left(x\right)\,A\right)$

$\left(x^2-1\right)f\left(x\right)+x\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)+x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}f\left(x\right)\right)$

$A≔\mathrm{SetOreRing}\left(n\,'\mathrm{shift}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(n\,\mathrm{shift}\right)$

$L≔\mathrm{OrePoly}\left(-\left(2n+1\right)\,2\left(n+1\right)\right)$

$L≔\mathrm{OrePoly}\left(-2n-1\,2n+2\right)$

$\mathrm{Apply}\left(L\,s\left(n\right)\,A\right)$

$\left(-2n-1\right)s\left(n\right)+\left(2n+2\right)s\left(n+1\right)$