The skew_product(w1, w2, A) function computes the noncommutative product $\mathrm{w1}\mathrm{w2}$ (in this order) in the Ore algebra A.

The skew_power(w, n, A) function computes the nth power of w in the Ore algebra A.

To declare an Ore algebra, use Ore_algebra[skew_algebra], Ore_algebra[shift_algebra], or Ore_algebra[diff_algebra].

These functions are part of the Ore_algebra package, and so can be used in the form skew_product(..) only after performing the command with(Ore_algebra) or with(Ore_algebra,skew_product).  The functions can always be accessed in the long form Ore_algebra[skew_product](..).

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

$A≔\mathrm{shift\_algebra}\left(\left[\mathrm{Sn}\,n\right]\right)\:$

$\mathrm{skew\_product}\left(\mathrm{Sn}\,n\,A\right)$

$\left(n+1\right)\mathrm{Sn}$

$\mathrm{skew\_product}\left({\mathrm{Sn}}^5\,n^3\,A\right)$

$\left(n^3+15n^2+75n+125\right){\mathrm{Sn}}^5$

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

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

$\mathrm{Dx}x+1,\mathrm{Dy}y+1$

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

$\mathrm{Dx}\mathrm{Dy}xy+\mathrm{Dx}x+\mathrm{Dy}y+1$

$\mathrm{skew\_product}\left(\mathrm{Dx}\,x^{10}\,A\right)$

$\mathrm{Dx}{x}^{10}+10x^9$

$\mathrm{skew\_power}\left(\mathrm{Dx}+x\,4\,A\right)$

${\mathrm{Dx}}^4+4x{\mathrm{Dx}}^3+\left(6x^2+6\right){\mathrm{Dx}}^2+\left(4x^3+12x\right)\mathrm{Dx}+x^4+6x^2+3$

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

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

$\mathrm{Dx}x+1,\left(n+1\right)\mathrm{Sn}$

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

$\left(xn+x\right)\mathrm{Dx}\mathrm{Sn}+\left(n+1\right)\mathrm{Sn}$

$\mathrm{skew\_product}\left(\mathrm{Sn}\,n^5\,A\right)$

$\left(n^5+5n^4+10n^3+10n^2+5n+1\right)\mathrm{Sn}$

$\mathrm{skew\_product}\left({\mathrm{Sn}}^5\,n\,A\right)$

$\left(n+5\right){\mathrm{Sn}}^5$