The SetOreRing(var, 'shift') calling sequence defines a shift algebra.

The SetOreRing([var, q], 'qshift') calling sequence defines a qshift algebra.

The SetOreRing(var, 'differential') calling sequence defines a differential algebra.

The shift, qshift, and differential algebras are pre-defined. You can use the SetOreRing command to define other Ore polynomial rings. You must specify procedures to compute sigma, sigma_inverse, and delta, and an expression to define theta(1).

For a brief review of pseudo-linear algebra (also known as Ore algebra), see OreAlgebra.

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

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

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

B := SetOreRing(n, 'difference',
  'sigma' = proc(p, x) eval(p, x=x+1) end,
  'sigma_inverse' = proc(p, x) eval(p, x=x-1) end,
  'delta' = proc(p, x) eval(p, x=x+1) - p end,
  'theta1' = 0);

$B≔\mathrm{UnivariateOreRing}\left(n\,\mathrm{difference}\right)$