Table of Contents

The procedure **HypergeometricSolution** in the package **LRS** (Linear Recurrence Systems) is an implementation in Maple 2015 of an algorithm to construct a general *hypergeometric* solutions for systems of linear recurrence equations with rational-function coefficients and hypergeometric right-hand sides.

Additionally, a procedure **RationalSolution** construct a general rational solutions for systems of linear recurrence equations with rational-function coefficients and rational-function right-hand sides.

The *matrix form* of a full-rank system:

**An(x) y(x+n) + … + A1(x) y(x+1) + A0(x) y(x) = b(x)**,

where

**Ai(x)**are matrices whose entries are rational functions of**x**;**b(x)**is zero or a column vector of finite sums of hypergeometric terms of**x**;**y(x)**is a column vector of unknown functions.

The *normal form* of a first order system:

**y(x+1) = A(x) y(x)**,

where

**A(x)**is a matrix whose entries are rational functions of**x**.

lrshypergeomsols.mpl - the Maple code of the package (implemented by A.A.Ryabenko in May, 2018).

lrs_exapmles_2018.mw - the Maple session file with examples.

lrs_exapmles_2018.pdf - the pdf copy of that Maple session.

The procedure **HypergeometricSolution** in the package **LRS** (Linear Recurrence Systems) is an implementation in Maple 2015 of an algorithm to construct a basis of the space of hypergeometric solutions for *homogeneous* systems in the forms above of linear recurrence equations with rational function coefficients.
The algorithm is based on the use of resolving sequences.

The details can be found in the paper S.A.Abramov, M.Petkovšek, A.A.Ryabenko. Resolving sequences of operators for linear ordinary differential and difference systems. Computational Mathematics and Mathematical Physics, 2016, Vol. 56, Issue. 5, P. 894–910.

Source:

lrshypergeomsols2016.mpl - the Maple code of the package (implemented by A.A.Ryabenko in 2016).

Paper_Examples_HS.mw - the Maple session file with examples from the paper.

Paper_Examples_HS.pdf - the pdf copy of that Maple session.

A procedure **HypergeometricSolution** in the package **LRS** is an implementation in Maple 18 for the algorithm of 2015 to find a basis of the space of hypergeometric solutions of a system **y(x+1) = A(x) y(x)**. Additionally, a procedure **Resolving** finds a resolving sequence of scalar equations and a resolving matrix for the system, a procedure **CyclicVector** finds an equivalent scalar equation for the system.

lrshypergeomsolscasc2015.mpl - the Maple code of the package;

hypergeometricsolution.mw - the Maple session file with examples of using the HypergeometricSolution procedure;

hypergeometricsolution.pdf - the pdf-copy of that Maple session;

slides.pdf - the slides of the talk S.A. Abramov, M. Petkovšek, A.A. Ryabenko "Hypergeometric Solutions of First-Order Linear Difference Systems with Rational-Function Coefficients" in The 18th workshop on computer algebra, May 26-27, 2015, Dubna

and of the talk S.A. Abramov, M. Petkovšek, A.A. Ryabenko "Hypergeometric Solutions of First-Order Linear Difference Systems with Rational-Function Coefficients" in The 17th international workshop on computer algebra in scientific computing, September 14-18, 2015, Aachen, Germany;

examples_casc2015.mw - the Maple session for samples in the CASC'2015 talk;

examples_casc2015.pdf - the pdf copy of that Maple session.