The HilbertTwistReduction(P, A, B) calling sequence returns the image of P under the Hilbert twist reduction.  If the (optional) third argument B is present, it is assigned to the Ore ring whose automorphism is the same as the A's and whose (pseudo) derivation sends everything to zero.

The InverseOfHilbertTwistReduction(P, A) calling sequence returns the pre-image of P under the Hilbert twist reduction. Note that A is the source ring of the Hilbert twist reduction.

Let A be the shift, q-shift, or differential algebra. The AccurateIntegration(L, A) calling sequence performs accurate integration, which solves the following problem: Let y satisfy L(y)=0 and g satisfy lambda(g)=y, where lambda means the usual derivative in the differential case, the difference operator in the shift case, and the q-difference operator in the q-shift case. The function builds an annihilator S (represented as an OrePoly structure) for g of the same degree as that of L, and an operator K such that g=K(y) if both exist. Otherwise, it returns $L\mathrm{\lambda }$.

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

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

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

A := 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);

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

$\mathrm{P1}≔\mathrm{OrePoly}\left(n\,n-1\right)$

$\mathrm{P1}≔\mathrm{OrePoly}\left(n\,n-1\right)$

$\mathrm{Q1}≔\mathrm{HilbertTwistReduction}\left(\mathrm{P1}\,A\,'B'\right)$

$\mathrm{Q1}≔\mathrm{OrePoly}\left(1\,n-1\right)$

$v≔\mathrm{GetVariable}\left(B\right)$

$v≔n$

$\mathrm{GetSigma}\left(B\right)\left(s\left(v\right)\,v\right)$

$s\left(n+1\right)$

$\mathrm{Getdelta}\left(B\right)\left(s\left(v\right)\right)$

$0$

$\mathrm{R1}≔\mathrm{InverseOfHilbertTwistReduction}\left(\mathrm{Q1}\,A\right)$

$\mathrm{R1}≔\mathrm{OrePoly}\left(n\,n-1\right)$

$A≔\mathrm{SetOreRing}\left(x\,'\mathrm{differential}'\right)$

$A≔\mathrm{UnivariateOreRing}\left(x\,\mathrm{differential}\right)$

$L≔\mathrm{OrePoly}\left(\frac{2\left(3x+1\right)}{x^2\left(4+27x\right)}\,-\frac{2}{x\left(4+27x\right)}\,1\right)$

$L≔\mathrm{OrePoly}\left(\frac{2\left(3x+1\right)}{x^2\left(4+27x\right)}\,-\frac{2}{x\left(4+27x\right)}\,1\right)$

$\mathrm{AccurateIntegration}\left(L\,A\right)$

$\left[\mathrm{OrePoly}\left(1\,-\frac{162x^2-9x-2}{180x}\,\frac{\left(3x-1\right)\left(4+27x\right)}{180}\right)\,\mathrm{OrePoly}\left(\frac{162x^2-9x-2}{180x}\,-\frac{\left(3x-1\right)\left(4+27x\right)}{180}\right)\right]$

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

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

$L≔\mathrm{OrePoly}\left(1\,-2\,-2\,1\right)$

$L≔\mathrm{OrePoly}\left(1\,\mathrm{-2}\,\mathrm{-2}\,1\right)$

$\mathrm{AccurateIntegration}\left(L\,A\right)$

$\left[\mathrm{OrePoly}\left(-\frac{1}{2}\,1\,1\,-\frac{1}{2}\right)\,\mathrm{OrePoly}\left(-\frac{3}{2}\,-\frac{1}{2}\,\frac{1}{2}\right)\right]$

Abramov, S.A., and van Hoeij, M. "Integration of Solutions of Linear Functional Equations." Integral Transformations and Special Functions. Vol. 8 No. 1-2. (1999): 3-12.