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

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

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

$\mathrm{Poly}≔\mathrm{OrePoly}\left(\frac{2\left(3x+1\right)}{x^2\left(4+27x\right)}\,-\frac{2}{x\left(4+27x\right)}\,1\right)$

$\mathrm{Poly}≔\mathrm{OrePoly}\left(\frac{2\left(3x+1\right)}{x^2\left(4+27x\right)}\,-\frac{2}{x\left(4+27x\right)}\,1\right)$

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

$\frac{2\left(3x+1\right)f\left(x\right)}{x^2\left(4+27x\right)}-\frac{2\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)}{x\left(4+27x\right)}+\frac{{\ⅆ}^2}{\ⅆx^2}f\left(x\right)$

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

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

$\mathrm{Poly}≔\mathrm{OrePoly}\left(1\,-2\,-2\,1\right)$

$\mathrm{Poly}≔\mathrm{OrePoly}\left(1\,\mathrm{-2}\,\mathrm{-2}\,1\right)$

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

$s\left(n\right)-2s\left(n+1\right)-2s\left(n+2\right)+s\left(n+3\right)$

$A≔\mathrm{SetOreRing}\left(\left[x\,q\right]\,'\mathrm{qshift}'\right)$

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

$\mathrm{Poly}≔\mathrm{OrePoly}\left(-q\left(1-qx\right)\,1\right)$

$\mathrm{Poly}≔\mathrm{OrePoly}\left(-q\left(-qx+1\right)\,1\right)$

$\mathrm{Apply}\left(\mathrm{Poly}\,s\left(x\right)\,A\right)$

$-q\left(-qx+1\right)s\left(x\right)+s\left(qx\right)$