Hypergeometric solutions for linear difference systems

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.

A system can be represented in several forms.

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.

Source

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 second version (only for homogeneous systems)

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.

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.

The first version (only for normal first order homogeneous systems)

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;

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

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