# Resolving sequences for systems of functional linear ordinary homogeneous equations with polynomial coefficients

The procedure ResolvingSequence is an implementation in Maple 2015 of an algorithm to construct resolving sequences for systems of differential, difference and q-difference equations with polynomial coefficients. The procedure is applicable to systems of equations from any Ore polynomial ring which can be defined by the procedure SetOreRing in the package OreTools (it is a standard package in Maple).

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. See Definition 2 of a resolving sequence in the Section 2.3 of the paper.

## A system can be represented in several forms.

The matrix form of a full-rank system:

An(x) ξ^n(y(x)) + … + A1(x) ξ(y(x)) + A0(x) y(x) = 0

where

• Ai(x) are matrices whose entries are polynomial of x;
• y(x) is a column vector of unknown functions;
• ξ is a pseudo-linear map

The OrePoly form of a full-rank system:

OrePoly(A0(x), A1(x), … An(x))

The normal form of a first order system:

ξ(y(x)) = A(x) y(x),

where

• A(x) is a matrix whose entries are rational functions in x.

For the differential case: ξ(y(x)) = y'(x).

For the difference case: ξ(y(x)) = y(x+1).

For the q-difference case: ξ(y(x)) = y(q x).

For any pseudo-linear map, ξ can be defined by the OreTools:-SetOreRing procedure (see ?help(“OreTools:-SetOreRing”) in Maple).

## Source

ResolvingSequence.mpl - the Maple code of the procedure (implemented by A.A.Ryabenko).

Paper_Examples.mw - the Maple session file with examples from 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.

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

samples2016.tar.gz - Maple sessions and their pdf copy for samples in the 19-th workshop talk.

samples2016_June.tar.gz - Maple sessions and their pdf copy for samples in the Conference “Computer Algebra”, Moscow, 2016.