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The LqRS (Linear q-Recurrence Systems) package implements in Maple 2017 a family of algorithms to solve a full-rank q-recurrence system.
We consider an arbitrary order linear system S of q-recurrence equations with polynomial coefficients:
An(x) y(x q^n) + … + A1(x) y(x q) + A0(x) y(x) = b(x),
where
The LqRS package is useful for solving the following types of problems:
The details can be found in the paper S.A. Abramov, A.A. Ryabenko, D.E. Khmelnov. Laurent, Rational, and Hypergeometric Solutions of Linear q-Difference Systems of Arbitrary Order with Polynomial Coefficients. Programming and Computer Software, 2018, Vol. 44, No. 2, pp. 120–130 (translation from С.А. Абрамов, А.А. Рябенко, Д.Е. Хмельнов. Лорановы, рациональные и гипергеометрические решения линейных q-разностных систем произвольного порядка с полиномиальными коэффициентами. Программирование, No 2, 2018, стр. 60-73).
The LqRS package works with systems which coefficients are rational functions of one variable, for example x, over ℚ(q), and right-hand sides are hypergeometric of x, over ℚ(q). (It's simple to transform them to systems with polynomial coefficients). ℚ(q) is an algebraic extension of the rational number field by q. The system S can be given in one of two alternative forms:
* the matrix-form:
* the list-form:
The procedure EG has the following arguments:
If the first argument is 'lead', the procedure EG constructs an l-embracing system, whose the leading matrix coefficient being invertible, and with the set of solutions containing all the solutions of S. If the first argument is 'trail', it constructs a t-embracing system, whose the trail matrix coefficient being invertible, and with the set of solutions containing all the solutions of S.
The output is a pair:
The procedure LaurentSolution has the following arguments:
The procedure LaurentSolution constructs the initial terms of the Laurent series solutions of S. The order of the initial terms may be greater than d if it is needed to reveal the full solution space dimension.
The output is a vector column which involves arbitrary constants of the form _c[1], _c[2], etc.
The procedure PolynomialSolution has the following arguments:
The procedure PolynomialSolution constructs the polynomial solutions of S. The output is a vector column which involves arbitrary constants of the form _c[1], _c[2], etc.
The procedure UniversalDenominator has the following arguments:
The procedure UniversalDenominator returns a universal denominator of S.
The procedure RationalSolution has the following arguments:
The procedure RationalSolution constructs the rational solutions of S. The output is a vector column which involves arbitrary constants of the form _c[1], _c[2], etc.
The procedure ResolvingSequence has the following arguments:
The procedure ResolvingSequence constructs a resolving sequence of S. See details in ResolvingSequence procedure. The output is a list of q-recurrence equations.
The procedure HypergeometricSolution has the following arguments:
The procedure HypergeometricSolution constructs the hypergeometric solutions of S. They are a finite linear combination of hypergeometric terms. A hypergeometric term h(x) is that the ratio of h(q x) and h(x) is a rational function in x.
The output is a vector column which involves arbitrary constants of the form _c[1], _c[2], etc.
lqrs.zip - the archive with two files lqrs.ind and lqrs.lib which are a Maple library. Put these files to some directory, for example, to ”/usr/userlib”. Assign libname := ”/usr/userlib”, libname in the Maple session. Then use the LqRS package.
paper_examples_lqrs.mw - the Maple session file with examples from the paper S.A. Abramov, A.A. Ryabenko, D.E. Khmelnov. Laurent, Rational, and Hypergeometric Solutions of Linear q-Difference Systems of Arbitrary Order with Polynomial Coefficients. Programming and Computer Software, 2018, Vol. 44, No. 2, pp. 120–130 .
paper_examples_lqrs.pdf - the pdf copy of that Maple session.
hyper_inhomogeneous.mw - the Maple session file with an example to construct hypergeometric solutions for inhomogeneous systems.
hyper_inhomogeneous.pdf - the pdf copy of that Maple session.