Here is an example of operators over a finite field.

$\mathrm{with}\left(\mathrm{Ore\_algebra}\right)\:$

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\right)\:$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x^5\,A\right)$

$\mathrm{Dx}{x}^5+5x^4$

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\,\mathrm{characteristic}=5\right)\:$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x^5\,A\right)$

$\mathrm{Dx}{x}^5$

Here are Ore algebras on a polynomial ring and on a rational function field.  The types of coefficients allowed differ accordingly.  In particular, generic functions are allowed in the rational case only, and have to be explicitly declared.

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\,\mathrm{polynom}=x\right)\:$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x\,A\right)$

$\mathrm{Dx}x+1$

On the other hand, both following inputs are illegal:

$\mathrm{skew\_product}\left(\mathrm{Dx}\,\frac{x}{x-1}\,A\right)$

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

$\mathrm{skew\_product}\left(\mathrm{Dx}\,\mathrm{\eta }\left(x\right)\,A\right)$

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\right)\:$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x\,A\right)$

$\mathrm{Dx}x+1$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,\frac{x}{x-1}\,A\right)$

$-\frac{1}{{\left(x-1\right)}^2}+\frac{\mathrm{Dx}x}{x-1}$

This is an error:

$\mathrm{skew\_product}\left(\mathrm{Dx}\,\mathrm{\eta }\left(x\right)\,A\right)$

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\,\mathrm{func}=\mathrm{\eta }\right)\:$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x\,A\right)$

$\mathrm{Dx}x+1$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,\frac{x}{x-1}\,A\right)$

$-\frac{1}{{\left(x-1\right)}^2}+\frac{\mathrm{Dx}x}{x-1}$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,\mathrm{\eta }\left(x\right)\,A\right)$

$\mathrm{Dx}\mathrm{\eta }\left(x\right)+\frac{\ⅆ}{\ⅆx}\mathrm{\eta }\left(x\right)$

This is an error:

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\,\mathrm{comm}=i\,\mathrm{alg\_relations}=i^2+1\right)\:$

$\mathrm{skew\_product}\left(i\mathrm{Dx}\,-i\mathrm{\eta }\left(x\right)\,A\right)$

Error, (in Ore_algebra:-skew_product) skew polynomials must be members of the algebra

This is not:

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\,\mathrm{comm}=i\,\mathrm{alg\_relations}=i^2+1\,\mathrm{func}=\mathrm{\eta }\right)\:$

$\mathrm{skew\_product}\left(i\mathrm{Dx}\,-i\mathrm{\eta }\left(x\right)\,A\right)$

$\mathrm{Dx}\mathrm{\eta }\left(x\right)+\frac{\ⅆ}{\ⅆx}\mathrm{\eta }\left(x\right)$

Each commutation type has its default action on Maple objects.  For instance, the diff commutation acts on functions f(x) and not on sequences u(n):

$A≔\mathrm{skew\_algebra}\left(\mathrm{diff}=\left[\mathrm{Dx}\,x\right]\right)\:$

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

$xf\left(x\right)$

$\mathrm{applyopr}\left(\mathrm{Dx}\,f\left(x\right)\,A\right)$

$\frac{\ⅆ}{\ⅆx}f\left(x\right)$

$\mathrm{applyopr}\left(x\,u\left(n\right)\,A\right)$

$xu\left(n\right)$

$\mathrm{applyopr}\left(\mathrm{Dx}\,u\left(n\right)\,A\right)$

$0$

By changing the action, you can view the previous Weyl algebra as acting on sequences u(n) rather than on functions f(x).

A:=skew_algebra(diff=[Dx,x],polynom=x,action={
    Dx=proc(u,order) local res; global n;
        res:=u; to order do res:=subs(n=n+1,n*res) end do; res
    end proc,
    x=proc(u,order) global n;
        subs(n=n-order,u)
    end proc}):

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

$f\left(x\right)$

$\mathrm{applyopr}\left(\mathrm{Dx}\,f\left(x\right)\,A\right)$

$\left(n+1\right)f\left(x\right)$

$\mathrm{applyopr}\left(x\,u\left(n\right)\,A\right)$

$u\left(n-1\right)$

$\mathrm{applyopr}\left(\mathrm{Dx}\,u\left(n\right)\,A\right)$

$\left(n+1\right)u\left(n+1\right)$