Overview of the QDifferenceEquations Package - Maple Help

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Overview of the QDifferenceEquations Package

 

Calling Sequence

Description

List of QDifferenceEquations Package Commands

Examples

References

Calling Sequence

QDifferenceEquations[command](arguments)

command(arguments)

Description

• 

The QDifferenceEquations package is useful for solving the following types of problems.

  

*   Finding polynomial solutions of a linear q-difference equation with polynomial coefficients.

  

*   Finding rational solutions of a linear q-difference equation with polynomial coefficients.

  

*   Finding q-hypergeometric solutions of a linear q-difference equation with polynomial coefficients.

  

*   Finding series solutions of a linear q-difference equation with polynomial coefficients.

  

*   Finding the universal denominator of the rational solutions of a linear q-difference equation with polynomial coefficients.

  

*   Computing the q-dispersion of two polynomials.

• 

For a given linear q-difference equation with polynomial coefficients, the main functionality of this package is to find the closed-form solution (namely, polynomial, rational, and q-hypergeometric) of the given equation. For finding the rational solution, the package constructs a universal denominator of such a solution. In turn the construction of the universal denominator is based on the computation of q-dispersion of two polynomials.

• 

The package also supports different basic q-hypergeometric objects, their simplification, expansion, and conversion.

• 

Each command in the QDifferenceEquations package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

  

As the underlying implementation of the QDifferenceEquations package is a module, it is also possible to use the form QDifferenceEquations:-command to access a command from the package. For more information,  see Module Members.

List of QDifferenceEquations Package Commands

• 

The following is a list of available commands.

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

 

• 

The PolynomialSolution, RationalSolution and SeriesSolution commands solve the problem with a single q-difference equation and also with a system of such equations. In the latter case the commands invoke LinearFunctionalSystems[PolynomialSolution], LinearFunctionalSystems[RationalSolution] and LinearFunctionalSystems[SeriesSolution] correspondingly in order to find solutions.

  

To display the help page for a particular QDifferenceEquations command, see Getting Help with a Command in a Package.

Examples

withQDifferenceEquations:

eq11q10qq10xyq2x1q20q2q20xyqx+q101q10q2q11xyx=q21q20q12+q10+q2qx

eq1:=1q10q10+qxyq2x1q20q20+q2xyqx+q101q10q11+q2xyx=q21q20q12+q10+q2qx

(1)

sol1PolynomialSolutioneq1,yx,,output=basis_K

sol1:=x10_K1+x_K2_K2+1

(2)

IsSolutionsol1,eq1,yx

true

(3)

eq2q3qx+1yq2x2q2x+1yqx+yxx+q=q62q3+1x2+xq52q3+q

eq2:=q3qx+1yq2x2q2x+1yqx+yxx+q=q62q3+1x2+xq52q3+q

(4)

sol2RationalSolutioneq2,yx,,output=anysol

sol2:=qx2+x3+y1q+y1q1xx+q

(5)

IsSolutionsol2,eq2,yx

true

(6)

eq34q2+qxq2yx+4q2+q5xyxq3+4q2x2yqx+2qx+2q2x2yxq2=xq32q2+5qx+2x2+2q3+2q5x+2q6x+2q8x2+2qx2+2x3q+3q2x2+2x3q3

eq3:=4q2+qxq2yx+4q5x+q2yq3x+4yqxq2x2+22q2x2+qxyq2x=xq32q8x2+2q6x+2q5x+2q3x3+3q2x2+2qx3+2q3+2qx2+2q2+5qx+2x2

(7)

sol3RationalSolutioneq3,yx,,output=basis_C

sol3:=12xx+q

(8)

IsSolutionsol3,eq3,yx

true

(9)

References

  

Abramov, S.A. "Problems in Computer Algebra Related to Constructing Solutions to Linear Difference Equations with Polynomial Coefficients." Vest. Mosk. Univ., Ser. 15. Vychisl. Mat. Kibern. No. 3. (1989): 56-60.

  

Abramov, S.A. "Rational Solutions to Linear Difference and q-Difference Equations with Polynomial Coefficients." Programmirovanie. No. 6. (1995):3-11.

  

Abramov, S.A.; Bronstein, M.; and Petkovsek, M. "On polynomial solutions of linear operator equations." Proceedings of ISSAC'95, pp. 290-296. ACM Press: New York, 1995.

  

Abramov, S.A.; Paule, P.; and Petkovsek, M. "q-Hypergeometric solutions of q-difference equations." Discrete Math. Vol. 180. (1998): 3-22.

  

Abramov, S.A., and Petkovsek, M. "Finding all q-hypergeometric solutions of q-difference equations." Proc. FPSAC '95, Univ.de Marne-la-Vall'ee, Noisy-le-Grand. pp. 1-10. 1995.

  

Boeing, H., and Koepf, W. "Algorithms for q-hypergeometric summation in computer algebra." Journal of Symbolic Computation. Vol. 11. (1999): 1-23.

  

Khmelnov, D.E. "Improved Algorithms for Solving Difference and q-Difference Equations." Programming and Computer Software Vol. 26 No. 2. (2000): 107-115. Translated from Programmirovanie. No. 2. 2000.

  

Man, Yiu-Kwong, and Wright, Francis J. "Fast Polynomial Dispersion Computation and its Application to Indefinite Summation." Proceedings of ISSAC'94, pp. 175-180. ACM Press: New York, 1994.

See Also

examples/QDifferenceEquations

LinearFunctionalSystems

module

UsingPackages

with

 


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