The Add(Ore1, ..., Orek) calling sequence adds the Ore polynomials Ore1,..., Orek.

The Minus(Ore1, Ore2) calling sequence subtracts the Ore polynomial Ore2 from the Ore polynomial Ore1.

The ScalarMultiply(s, Ore1) calling sequence multiplies the Ore polynomial Ore1 on the left by the scalar s.

The Multiply(Ore1, ..., Orek, A) calling sequence multiplies the t Ore polynomials Ore1, ...,  Orek in the Ore algebra A.

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

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

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

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

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

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

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

$\mathrm{Add}\left(\mathrm{Ore1}\,\mathrm{Ore2}\,\mathrm{Ore1}\right)$

$\mathrm{OrePoly}\left(-\frac{n\left(n+1\right)}{n-1}\,-\frac{n^3-2n^2-6n+5}{n-1}\,n^2-5\right)$

$\mathrm{Minus}\left(\mathrm{Ore1}\,\mathrm{Ore2}\right)$

$\mathrm{OrePoly}\left(\frac{n\left(-2+n\right)}{n-1}\,\frac{n^3-5n^2+9n-4}{n-1}\,-n^2+3n-4\right)$

$\mathrm{ScalarMultiply}\left(\mathrm{sqrt}\left(2\right)\,\mathrm{Ore1}\right)$

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

$\mathrm{Multiply}\left(\mathrm{Ore1}\,\mathrm{Ore2}\,\mathrm{Ore1}\,A\right)$

$\mathrm{OrePoly}\left(-\frac{n^3}{{\left(n-1\right)}^2}\,-\frac{3n^5-12n^4+10n^2+n-3}{n{\left(n-1\right)}^2}\,-\frac{3n^6-18n^5+2n^4+62n^3+8n^2-29n-3}{n\left(n-1\right)\left(n+1\right)}\,-\frac{n^7-11n^6+3n^5+92n^4-19n^3-168n^2+8n+75}{\left(n-1\right)\left(n+1\right)\left(n+2\right)}\,\frac{3n^7-5n^6-55n^5+37n^4+209n^3-100n^2-142n+57}{\left(n-1\right)\left(n+2\right)\left(n+3\right)}\,-\frac{3n^6-n^5-34n^4+2n^3+31n^2-2n-3}{\left(n-1\right)\left(n+3\right)}\,\left(n-3\right){\left(n+1\right)}^3\right)$