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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

Parameters

Description

Examples

References

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 Lλ.

Examples

withOreTools:

withOreTools[MathOperations]:

withOreTools[Properties]:

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);

AUnivariateOreRingn,difference

(1)

P1OrePolyn,n1

P1OrePolyn,n1

(2)

Q1HilbertTwistReductionP1,A,'B'

Q1OrePoly1,n1

(3)

vGetVariableB

vn

(4)

GetSigmaBsv,v

sn+1

(5)

GetdeltaBsv

0

(6)

R1InverseOfHilbertTwistReductionQ1,A

R1OrePolyn,n1

(7)

Examples of AccurateIntegration:

ASetOreRingx,'differential'

AUnivariateOreRingx,differential

(8)

LOrePoly23x+1x24+27x,2x4+27x,1

LOrePoly23x+1x24+27x,2x4+27x,1

(9)

AccurateIntegrationL,A

OrePoly1,1180162x29x2x,11803x14+27x,OrePoly1180162x29x2x,11803x14+27x

(10)

ASetOreRingn,'shift'

AUnivariateOreRingn,shift

(11)

LOrePoly1,2,2,1

LOrePoly1,2,2,1

(12)

AccurateIntegrationL,A

OrePoly12,1,1,12,OrePoly32,12,12

(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.

See Also

OreTools

OreTools/OreAlgebra

OreTools/OrePoly

OreTools[MathOperations]

OreTools[Properties]

OreTools[SetOreRing]

 


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