Solutions for linear q-recurrence systems with polynomial coefficients

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

• Ai(x) are matrices whose entries are rational functions of x;
• b(x) is 0 or a column vector whose entries are hypergeometric of x;
• y(x) is a column vector of unknown functions;
• q is a name.

The LqRS package is useful for solving the following types of problems:

• Transform S into an embracing system with a non-singular leading or trailing matrix. These constructions are performed by using the EG-elimination algorithm (The procedure EG).
• Find Laurent series solutions of S (The procedure LaurentSolution).
• Find polynomial solutions of S (The procedure PolynomialSolution).
• Find the universal denominator of the rational solutions of S (The procedure UniversalDenominator).
• Find rational solutions of S (The procedure RationalSolution).
• Construct a resolving sequence of S, which is homogeneous (The procedure ResolvingSequence).
• Find hypergeometric solutions of S (The procedure HypergeometricSolution).

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:

1. a q-recurrence equation for one unknown with matrix coefficients and a vector right-hand side,
2. an unknown, for example y(x);

* the list-form:

1. a list (or a set) of q-recurrence equations for several unknowns with rational functions coefficients and hypergeometric righ-hand sides,
2. a list of unknowns, for example [y1(x),…, ym(x)].

The procedure EG has the following arguments:

1. the word 'lead' or the word 'trail',
2. a system S (see System representations).

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:

1. the system in the same form as the given S,
2. true if S has a full rank, and false otherwise.

The procedure LaurentSolution has the following arguments:

1. a system S which right hand side is 0 or a column vector whose entries are rational functions of x (see System representations),
2. (optional argument) an order d of the initial terms of the solution series to calculate.

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:

1. a system S (see System representations).

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:

1. a system S (see System representations).

The procedure UniversalDenominator returns a universal denominator of S.

The procedure RationalSolution has the following arguments:

1. a system S (see System representations).

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:

1. a homogeneous system S (see System representations),
2. (optional argument) 'select_indicator' = i where i is a number of an unknown function which have to be first in a constructed resolving sequence.

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:

1. a system S (see System representations),
2. a name k to construct hypergeometric solutions.

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.

12_00_ab_slides11_10_2017.pdf - the slides of the talk Sergei Abramov EG-eliminations as a tool for computing rational solutions of linear q-difference systems of arbitrary order with polynomial coefficients in 2nd International Conference "Computer Algebra", Moscow, October 30 - November 3, 2017.

11_30_ryabenko.pdf - the slides of the talk Anna Ryabenko Search for hypergeometric solutions of q-difference systems by resolving sequences in 2nd International Conference "Computer Algebra", Moscow, October 30 - November 3, 2017.

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.