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Q-Difference Equations

 

The QDifferenceEquations package provides tools for studying equations of the form:

 

Lyx=anxyqnx+an1xyqn1x++a1xyq x+a0xyx=0,

 

and their solutions yx, where a0, ... ,an are polynomials in the indeterminates x and q. The indeterminate q is considered to be a constant. L=anQn+an1Qn1+  +a1Q+a0 is the associated q-difference operator of order n, where Q represents the q-shift operator Qyx=yq x.

 

For example, the solutions of the first order q-difference equation Ly=0, where L=x21Qq2 x21, are given by:

 

yx=Cx21,

 

where C is an arbitrary constant that is allowed to depend on q, but not on x.

 

In Maple 18, two new commands were added to this package:

• 

Closure computes the closure in the ring of linear q-difference operators with polynomial coefficients.

• 

Desingularize computes a multiple of a given q-difference operator with fewer singularities.

 

As an example, let's look at the operator L from above.

withQDifferenceEquations

AccurateQSummation,AreSameSolution,Closure,Desingularize,ExtendSeries,IsQHypergeometricTerm,IsSolution,PolynomialSolution,QBinomial,QBrackets,QDispersion,QECreate,QEfficientRepresentation,QFactorial,QGAMMA,QHypergeometricSolution,QMultiplicativeDecomposition,QPochhammer,QPolynomialNormalForm,QRationalCanonicalForm,QSimpComb,QSimplify,RationalSolution,RegularQPochhammerForm,SeriesSolution,UniversalDenominator,Zeilberger

(1)

Lx21Qq2x21:

 

This operator has singularities at x=±1, where its leading coefficient vanishes. However, the solutions yx=Cx21 satisfying Ly=0 are non-singular at both points, so x=±1 are two apparent singularities. It is possible to remove such apparent singularities by finding a higher order operator M that has the same solutions as L, plus some additional ones. This is what the command Desingularize does.

 

MDesingularizeL,Q,x,q

M:=Q2+q21Q+q2

(2)

 

Let us verify that yx is actually a solution of M.

withOreTools:withConverters:

ASetOreRingx,q,qshift:

OMFromPolyToOrePolyM,Q

OM:=OrePolyq2,q21,1

(3)

ApplyOM,yx,A;

q2yx+q21yqx+yq2x

(4)

expandeval,y=xCx21

0

(5)

 

The closure of an operator L consists of all left "pseudo"-multiples of L, i.e., all operators R for which there exists an operator, P (in Q,x,q) and a polynomial f (in x,q only), such that the following torsion relation holds true:

PL=f R

Basically, this means that PL is a genuine left multiple of L of which one can factor out the content f. Both PL and R have exactly the same solutions, which include all solutions of L. In particular, the desingularizing operator M from above is an element of the closure of L.

 

The command Closure computes a basis of the closure.

CClosureL,Q,x,q

C:=x21Qq2x2+1,qx1Q2+q3x+q2+qx+1Qqx+1q2,qx1Q2+q3x+q2qx+1Q+qx1q2

(6)

 

We see that, trivially, L itself belongs to its closure. In addition, the basis contains two second order operators, both of which have fewer and different singularities than L itself, namely, x=q1 and x=q1, respectively. Since these two singularities are different, the two leading coefficients are coprime as polynomials in x, and we can find a linear combination that is monic:

gcdexlcoeffC2,Q,lcoeffC3,Q,x, 's', 't',s,t;

1,12,12

(7)

collectsC2+tC3,Q,normal

Q2+q21Q+q2

(8)

 

This, in fact, is exactly the desingularizing operator from above.


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