construct a desingularizing q-shift operator with polynomial coefficients - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Discrete Mathematics : Summation and Difference Equations : QDifferenceEquations : QDifferenceEquations/Desingularize

QDifferenceEquations[Desingularize] - construct a desingularizing q-shift operator with polynomial coefficients

Calling Sequence

Desingularize(L, Qx, x, q, func, options)

Parameters

L

-

polynomial in Qx with coefficients which are polynomials in x over the field of rational functions in q

Qx

-

name, variable denoting the q-shift operator xqx 

x

-

variable name

q

-

either a variable name, or a nonzero constant that is not a root of unity, or an equation of the form name=constant

func

-

(optional) procedure

options

-

(optional) equation of the form 'coeff'=t, where t is one of leading, trailing, or both

Returns

• 

polynomial in Qx with coefficients which are polynomials in x over the field of rational functions in q, which maximally desingularizes L

Description

• 

Let k be a field of characteristic 0. Denote by F the q-shift polynomial ring consisting of elements, each of which is a polynomial in Qx, with coefficients which are polynomials in x over kq. For a given operator LF, the Desingularize(L,Qx,x,q) calling sequence constructs an operator RF that maximally desingularizes the leading coefficient, the trailing coefficient, or both coefficients of L, depending on the option coeff. Equivalently, all apparent singularities of the leading coefficient, the trailing coefficient, or both coefficients of L are removed in R.

  

Note that R is right divisible by L over the field kq,x.

• 

The parameter q does not have to be a variable.  A nonzero constant value, such as, q=2 is possible as well; provided that it is not a root of unity, and thus satisfies qn1 for all positive integers n.

• 

The optional argument func, if specified, is applied to the coefficients of the result with respect to Qx; typical examples are expand or factor.

• 

Note that setting infolevel[Desingularize]:=3 will cause some diagnostics to be printed during the computation.

Options

• 

'coeff'=t, where t is one of leading, trailing, or both

  

Indicates whether the desingularization is done with respect to the leading coefficient, the trailing coefficient, or both coefficients of the input operator L. The default is leading.

Examples

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)

For the following q-shift operator L, compute desingularizing operators with respect to the leading coefficient and the trailing coefficient when q=13:

L:=x3qx3Qx+q2x32q3x3

L:=x3qx3Qx+q2x32q3x3

(2)

DesingularizeL,Qx,x,q=13,'coeff'='leading',factor

Qx33380729Qx2+22249764782969x+2293408531441Qx+110460353203x81x2+6703830x230132907

(3)

DesingularizeL,Qx,x,q=13,'coeff'='trailing',factor

832177147x8322187+45986561x406243Qx2Qx3+5111594323x221100177147x+927676561Qx

(4)

Note that in the latter case, not all singularities of the trailing coefficient could be removed; the factor q3x3=127x81 remains.

The following call returns an error since q=1 is a second root of unity:

DesingularizeL,Qx,x,q=1

Error, (in QDifferenceEquations:-Desingularize) unable to compute a desingularizing operator for the case where q is a root of unity

See Also

Groebner, Ore_algebra, QDifferenceEquations[Closure]


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam