The procedure HypergeometricSolution in the package LRS (Linear Recurrence Systems) is an implementation in Maple 2018 of an algorithm to construct a general hypergeometric solutions for systems of linear recurrence equations with rational-function coefficients and hypergeometric right-hand sides.

The details can be found in the paper A.A. Ryabenko. Particular solutions of linear differential and (q-) difference systems with hypergeometric right-hand sides // Programming and computer software. 2019. Vol. 45. No. 5. P. 298--302.

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 in x.

lrshypergeomsols.mpl - the Maple code of the package (implemented by A.A.Ryabenko).

lrs_exapmles_2018.mw - the Maple session file with examples to construct hypergeometric solutions for homogeneous and inhomogeneous systems.

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 of arbitrary order. Computational Mathematics and Mathematical Physics, 2016, Vol. 56, Issue. 5, P. 894–910.

Source:

lrshypergeomsols.mpl - the Maple code of the package (implemented by A.A.Ryabenko).

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.