The Modular[Add](Ore1, Ore2, m) calling sequence adds the two Ore polynomials Ore1 and Ore2 modulo p.

The Modular[Minus](Ore1, Ore2, p) calling sequence subtracts the Ore polynomial Ore2 from the Ore polynomial Ore1 modulo p.

The Modular[ScalarMultiply](s, Ore1, p) calling sequence multiplies the Ore polynomial Ore1 on the left by the scalar s modulo p.

The Modular[Multiply](Ore1, Ore2, p, A) calling sequence multiplies the two Ore polynomials Ore1 and Ore2 in the Ore algebra A modulo m.

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

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

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

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(-\frac{n}{n-1}\,-\frac{-5n+n^2+3}{n-1}\,n-3\right)$

$\mathrm{Ore1}≔\mathrm{OrePoly}\left(-\frac{n}{n-1}\,-\frac{n^2-5n+3}{n-1}\,n-3\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(-n\,3n-n^2-1\,{\left(n-1\right)}^2\right)$

$\mathrm{Ore2}≔\mathrm{OrePoly}\left(-n\,-n^2+3n-1\,{\left(n-1\right)}^2\right)$

$\mathrm{Modular}\left[\mathrm{Add}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,7\right)$

$\mathrm{OrePoly}\left(\frac{6n^2}{n+6}\,\frac{6n^3+3n^2+n+5}{n+6}\,n^2+6n+5\right)$

$\mathrm{Modular}\left[\mathrm{Minus}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,7\right)$

$\mathrm{OrePoly}\left(\frac{n^2+5n}{n+6}\,\frac{n^3+2n^2+2n+3}{n+6}\,6n^2+3n+3\right)$

$\mathrm{Modular}\left[\mathrm{ScalarMultiply}\right]\left(22\,\mathrm{Ore1}\,17\right)$

$\mathrm{OrePoly}\left(\frac{12n}{n+16}\,\frac{12n^2+8n+2}{n+16}\,5n+2\right)$

$\mathrm{Modular}\left[\mathrm{Multiply}\right]\left(\mathrm{Ore1}\,\mathrm{Ore2}\,11\,A\right)$

$\mathrm{OrePoly}\left(\frac{n^2}{n+10}\,\frac{2n^3+4n^2+10n+3}{n+10}\,\frac{n^4+3n^3+6n+2}{n+10}\,\frac{9n^4+8n^3+10n^2+4n+3}{n+10}\,\left(n+8\right){\left(n+1\right)}^2\right)$