The Ore_to_diff command converts a differential operator or a list of differential operators of the skew algebra A into a differential equation or a list of differential equations in the function f.  The output is expressed in terms of the diff function by default, or in terms of the D function when the optional parameter is set.

The Ore_to_DESol command converts a single differential operator of the skew algebra A into a DESol structure in the function f.

The Ore_to_shift command converts a shift operator or a list of shift operators of the skew algebra A into a recurrence equation or a list of recurrence equations in the sequence u.  The output is expressed in functional notation ( $u\left(n\right),...$ ) by default, or in the indexed notation ( $u_n,...$ ) when the optional argument is set.

The Ore_to_RESol command converts a single recurrence operator of the skew algebra A into an RESol structure in the sequence u.

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

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

$P≔x^2{\mathrm{Dx}}^2+x\mathrm{Dx}+x^2-{\mathrm{\mu }}^2\:$

$\mathrm{Ore\_to\_diff}\left(P\,f\,A\right)$

$x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}f\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)+\left(-{\mathrm{\mu }}^2+x^2\right)f\left(x\right)$

$\mathrm{Ore\_to\_diff}\left(P\,f\,A\,\mathrm{D}\right)$

$x^2{\mathrm{D}}^{\left(2\right)}\left(f\right)\left(x\right)+x\mathrm{D}\left(f\right)\left(x\right)+\left(-{\mathrm{\mu }}^2+x^2\right)f\left(x\right)$

$\mathrm{Ore\_to\_DESol}\left(P\,f\,A\right)$

$\mathrm{DESol}\left(\left\{x^2\left(\frac{{\ⅆ}^2}{\ⅆx^2}f\left(x\right)\right)+x\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)\right)+\left(-{\mathrm{\mu }}^2+x^2\right)f\left(x\right)\right\}\,\left\{f\left(x\right)\right\}\right)$

$\mathrm{normal}\left(\mathrm{applyopr}\left(P\,\,A\right)\right)$

$0$

$A≔\mathrm{skew\_algebra}\left(\mathrm{euler}=\left[\mathrm{Tx}\,x\right]\,\mathrm{comm}=\mathrm{\mu }\right)\:$

$P≔{\mathrm{Tx}}^2+x^2-{\mathrm{\mu }}^2\:$

$\mathrm{Ore\_to\_diff}\left(P\,f\,A\right)$

$x\left(\frac{\ⅆ}{\ⅆx}f\left(x\right)+x\left(\frac{{\ⅆ}^2}{\ⅆx^2}f\left(x\right)\right)\right)+\left(-{\mathrm{\mu }}^2+x^2\right)f\left(x\right)$

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

$P≔{\mathrm{Sn}}^2+\mathrm{\alpha }\mathrm{Sn}+1\:$

$\mathrm{Ore\_to\_shift}\left(P\,u\,A\right)$

$u\left(n+2\right)+\mathrm{\alpha }u\left(n+1\right)+u\left(n\right)$

$\mathrm{Ore\_to\_shift}\left(P\,u\,A\,\mathrm{indexed}\right)$

$\mathrm{\alpha }{u}_{n+1}+u_n+u_{n+2}$

$\mathrm{Ore\_to\_RESol}\left(P\,u\,A\right)$

$\mathrm{RESol}\left(\left\{u\left(n+2\right)+\mathrm{\alpha }u\left(n+1\right)+u\left(n\right)=0\right\}\,\left\{u\left(n\right)\right\}\,\left\{u\left(0\right)=u\left(0\right)\,u\left(1\right)=u\left(1\right)\right\}\,\mathrm{INFO}\right)$

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

$G≔\left[-2\mathrm{Dx}x+\mathrm{Dy}y\,-4\left(\mathrm{\mu }-x{y}^2\right)\left(\mathrm{\mu }+x{y}^2\right)+2\mathrm{Dx}x+y^2{\mathrm{Dy}}^2\,-2\left(\mathrm{\mu }-x{y}^2\right)\left(\mathrm{\mu }+x{y}^2\right)+\mathrm{Dy}\mathrm{Dx}yx\,-\left(\mathrm{\mu }-x{y}^2\right)\left(\mathrm{\mu }+x{y}^2\right)+\mathrm{Dx}x+{\mathrm{Dx}}^2{x}^2\right]\:$

$\mathrm{Ore\_to\_diff}\left(G\,f\,A\right)$

$\left[-2x\left(\frac{\partial }{\partial x}f\left(x\,y\right)\right)+y\left(\frac{\partial }{\partial y}f\left(x\,y\right)\right)\,-4\left(-x{y}^2+\mathrm{\mu }\right)\left(x{y}^2+\mathrm{\mu }\right)f\left(x\,y\right)+y^2\left(\frac{{\partial }^2}{\partial y^2}f\left(x\,y\right)\right)+2x\left(\frac{\partial }{\partial x}f\left(x\,y\right)\right)\,-2\left(-x{y}^2+\mathrm{\mu }\right)\left(x{y}^2+\mathrm{\mu }\right)f\left(x\,y\right)+yx\left(\frac{{\partial }^2}{\partial x\partial y}f\left(x\,y\right)\right)\,-\left(-x{y}^2+\mathrm{\mu }\right)\left(x{y}^2+\mathrm{\mu }\right)f\left(x\,y\right)+x\left(\frac{\partial }{\partial x}f\left(x\,y\right)\right)+x^2\left(\frac{{\partial }^2}{\partial x^2}f\left(x\,y\right)\right)\right]$

$\mathrm{Ore\_to\_diff}\left(G\,f\,A\,\mathrm{D}\right)$

$\left[-2x{\mathrm{D}}_1\left(f\right)\left(x\,y\right)+y{\mathrm{D}}_2\left(f\right)\left(x\,y\right)\,-4\left(-x{y}^2+\mathrm{\mu }\right)\left(x{y}^2+\mathrm{\mu }\right)f\left(x\,y\right)+y^2{\mathrm{D}}_{2,2}\left(f\right)\left(x\,y\right)+2x{\mathrm{D}}_1\left(f\right)\left(x\,y\right)\,-2\left(-x{y}^2+\mathrm{\mu }\right)\left(x{y}^2+\mathrm{\mu }\right)f\left(x\,y\right)+yx{\mathrm{D}}_{1,2}\left(f\right)\left(x\,y\right)\,-\left(-x{y}^2+\mathrm{\mu }\right)\left(x{y}^2+\mathrm{\mu }\right)f\left(x\,y\right)+x{\mathrm{D}}_1\left(f\right)\left(x\,y\right)+x^2{\mathrm{D}}_{1,1}\left(f\right)\left(x\,y\right)\right]$

$\mathrm{Ore\_to\_DESol}\left(G\,f\,A\right)$

Error, invalid input: Ore_algebra:-Ore_to_DESol expects its 1st argument, poly, to be of type polynom, but received [-2*Dx*x+Dy*y, -4*(-x*y^2+mu)*(x*y^2+mu)+2*Dx*x+y^2*Dy^2, -2*(-x*y^2+mu)*(x*y^2+mu)+Dy*Dx*y*x, -(-x*y^2+mu)*(x*y^2+mu)+Dx*x+Dx^2*x^2]