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QDifferenceEquations[AccurateQSummation] - sum the solutions of a q-shift operator

Calling Sequence

AccurateQSummation(L, Q, x)

Parameters

L

-

polynomial in Q over C(q)(x)

Q

-

name; denote the q-shift operator

x

-

name (that Q acts on)

Description

• 

This AccurateQSummation(L,Q,x) calling sequence computes an operator M of minimal order such that any solution f of L has an anti-qdifference which is a solution of M.

• 

If the order of L equals the order of M then the output is a list [M, r] such that r(f) is an anti-qdifference of f and also a solution of M for every solution f of L. If the order of L is not equal to M then only M is given in the output. In this case M equals LΔ where Δ=Q1.

• 

Q is the q-shift operator with respect to x, defined by Qx=qx.

Examples

withQDifferenceEquations:

L:=q1+q2Q2+q2q41Qq51+q2

L:=qq21Q2+q2q41Qq5q21

(1)

Ac:=AccurateQSummationL,Q,x

Ac:=q4q4q3q+1q2+1qQq4q3q+1+Q2q4q3q+1,q3+q1q4q3q+1Qq4q3q+1

(2)

Lt:=op1,Ac;rt:=op2,Ac

Lt:=q4q4q3q+1q2+1qQq4q3q+1+Q2q4q3q+1

rt:=q3+q1q4q3q+1Qq4q3q+1

(3)

Regarding the meaning of the second element rt in the output of AccurateQSummation, since L is the minimal annihilator of f=qx3+x, g=rtf is an anti-qdifference of f:

A:=OreTools:-SetOreRingx,q,'qshift':

f:=qx3+x

f:=qx3+x

(4)

r:=OreTools:-Converters:-FromPolyToOrePolyrt,Q:

g:=normalOreTools:-Applyr,f,A

g:=qx2+q2+q+1xq31

(5)

check that Q1g=f:

normalgx=qx|gx=qxgf

0

(6)

See Also

DEtools/integrate_sols, OreTools[Converters][FromPolyToOrePoly], OreTools[MathOperations][AccurateIntegration], OreTools[SetOreRing], SumTools[IndefiniteSum][AccurateSummation]

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.


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