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OreTools[MathOperations]

 HilbertTwistReduction
 return isomorphic images of Ore polynomials under the Hilbert twist reduction
 InverseOfHilbertTwistReduction
 return pre-images of Ore polynomials under the HilbertTwistReduction
 AccurateIntegration
 check for the existence of a primitive element, and perform accurate integration

 Calling Sequence HilbertTwistReduction(P,  A, 'B') InverseOfHilbertTwistReduction(P, A) AccurateIntegration(L, A) Note: An Ore polynomial ring B is of Hilbert's twist type if its (pseudo) derivation maps everything to zero. For an Ore polynomial ring A with nontrivial automorphism, there is a ring isomorphism from A onto the ring B of Hilbert's twist type whose automorphism is the same as the A's. The isomorphism is called the Hilbert twist reduction.

Parameters

 P - Ore polynomial or a list of Ore polynomials; to define an Ore polynomial, use the OrePoly structure. L - Ore polynomial. A - Ore ring with nontrivial automorphism; to define an Ore algebra, use the SetOreRing function. B - (optional) unevaluated name.

Description

 • 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 $L\mathrm{\lambda }$.

Examples

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

Define an Ore ring.

 > 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);
 ${A}{:=}{\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{difference}}\right)$ (1)
 > $\mathrm{P1}≔\mathrm{OrePoly}\left(n,n-1\right)$
 ${\mathrm{P1}}{:=}{\mathrm{OrePoly}}{}\left({n}{,}{n}{-}{1}\right)$ (2)
 > $\mathrm{Q1}≔\mathrm{HilbertTwistReduction}\left(\mathrm{P1},A,'B'\right)$
 ${\mathrm{Q1}}{:=}{\mathrm{OrePoly}}{}\left({1}{,}{n}{-}{1}\right)$ (3)
 > $v≔\mathrm{GetVariable}\left(B\right)$
 ${v}{:=}{n}$ (4)
 > $\mathrm{GetSigma}\left(B\right)\left(s\left(v\right),v\right)$
 ${s}{}\left({n}{+}{1}\right)$ (5)
 > $\mathrm{Getdelta}\left(B\right)\left(s\left(v\right)\right)$
 ${0}$ (6)
 > $\mathrm{R1}≔\mathrm{InverseOfHilbertTwistReduction}\left(\mathrm{Q1},A\right)$
 ${\mathrm{R1}}{:=}{\mathrm{OrePoly}}{}\left({n}{,}{n}{-}{1}\right)$ (7)

Examples of AccurateIntegration:

 > $A≔\mathrm{SetOreRing}\left(x,'\mathrm{differential}'\right)$
 ${A}{:=}{\mathrm{UnivariateOreRing}}{}\left({x}{,}{\mathrm{differential}}\right)$ (8)
 > $L≔\mathrm{OrePoly}\left(\frac{2\left(3x+1\right)}{{x}^{2}\left(4+27x\right)},-\frac{2}{x\left(4+27x\right)},1\right)$
 ${L}{:=}{\mathrm{OrePoly}}{}\left(\frac{{2}{}\left({3}{}{x}{+}{1}\right)}{{{x}}^{{2}}{}\left({4}{+}{27}{}{x}\right)}{,}{-}\frac{{2}}{{x}{}\left({4}{+}{27}{}{x}\right)}{,}{1}\right)$ (9)
 > $\mathrm{AccurateIntegration}\left(L,A\right)$
 $\left[{\mathrm{OrePoly}}{}\left({1}{,}{-}\frac{{1}}{{180}}{}\frac{{162}{}{{x}}^{{2}}{-}{9}{}{x}{-}{2}}{{x}}{,}\frac{{1}}{{180}}{}\left({3}{}{x}{-}{1}\right){}\left({4}{+}{27}{}{x}\right)\right){,}{\mathrm{OrePoly}}{}\left(\frac{{1}}{{180}}{}\frac{{162}{}{{x}}^{{2}}{-}{9}{}{x}{-}{2}}{{x}}{,}{-}\frac{{1}}{{180}}{}\left({3}{}{x}{-}{1}\right){}\left({4}{+}{27}{}{x}\right)\right)\right]$ (10)
 > $A≔\mathrm{SetOreRing}\left(n,'\mathrm{shift}'\right)$
 ${A}{:=}{\mathrm{UnivariateOreRing}}{}\left({n}{,}{\mathrm{shift}}\right)$ (11)
 > $L≔\mathrm{OrePoly}\left(1,-2,-2,1\right)$
 ${L}{:=}{\mathrm{OrePoly}}{}\left({1}{,}{-}{2}{,}{-}{2}{,}{1}\right)$ (12)
 > $\mathrm{AccurateIntegration}\left(L,A\right)$
 $\left[{\mathrm{OrePoly}}{}\left({-}\frac{{1}}{{2}}{,}{1}{,}{1}{,}{-}\frac{{1}}{{2}}\right){,}{\mathrm{OrePoly}}{}\left({-}\frac{{3}}{{2}}{,}{-}\frac{{1}}{{2}}{,}\frac{{1}}{{2}}\right)\right]$ (13)

References

 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.