# EG package

The package is implementing the family of EG algorithms.

## EG eliminations

We consider arbitrary order ordinary systems of differential or difference equations with polynomial coefficients. We assume that equations of the system are independent.

For any system S of this form the algorithm EG_delta in the differential case and the algorithm EG_sigma in the difference case construct an l-embracing system S' of the same form, but with the leading matrix coefficient being invertible, and with the set of solutions containing all the solutions of S.

The difference case is even more pliable. The algorithm EG_sigma can also construct a t-embracing system S'' of the same form, but with the trailing matrix coefficien being invertible, and with the set of solutions containing all the solutions of the system S. Additionally in the difference case, one can consider the system together with a finite set of linear constraints, i.e. linear relations, each of which contains a finite set of varlues of the indfinitite functions of the system at some points. Let S and S' be systems of the considering form in the difference case and C and C' be finite sets of linear constraints. We then say that the systems (S,C) and (S',C') are equivalent if the space of solutions of S that satisfy C is the same as the space of solutions of S' that satisfy C'. Each of l- and t-versions of EG_sigma computes a system (S',C') equivalent to (S,{}) and such that the leading or, resp., trailing matrix of S' is invertible.

EG_delta and EG_sigma are used for finding solutions of the considering systems and for finding and investigating singular points of analytic solutions of such systems.

Given a differential system, the Singsys procedure constructs a nonzero polynomial d(x) such that if the given system has Laurent series solution with in (x-a) for some point a, and a component of this solution has a nonzero polar part then d(a)=0. The procedure uses EG_delta algorithm.

## Solutions of differential systems with power series coefficients

Using EG_delta and EG_sigma algorithms, new algorithms were implemented: to find all Laurent, all regular and some formal exponential-logarithmic solutions for a linear differential system with power series coefficients.

For the cases Laurent and regular solutions, the details are to be available in the paper S.A.Abramov,D.E.Khmelnov. Regular Solutions of Linear Differential Systems with Power Series Coefficients. Programming and Computer Software, No 2, 2014, 98-106 (С.А.Абрамов, Д.Е.Хмельнов. Регулярные решения линейных дифференциальных систем с коэффициентами в виде степенных рядов. Программирование, No 2, 2014, стр. 75-85 ).

For the case of formal exponential-logarithmic solutions, the details are to be available in the paper A.A.Ryabenko. On Exponential–Logarithmic Solutions of Linear Differential Systems with Power Series Coefficients, 2015, No 2, pp. 112-118 (А.А.Рябенко. Экспоненциально-логарифмические решения линейных дифференциальных систем с коэффициентами в виде степенных рядов. Программирование 2015, No 2. С. 54–62), a Maple session for samples in this paper.

## Source

EG.mpl - the Maple code of the EG package.

EG.mws - the Maple session file with examples of using the procedures EG_delta, EG_sigma, Singsys.

LaurentRegularFormalSol.mw - the Maple session file with an example of find formal (exponetial-logarithmic) solutions, regular and Laurent solutions, when the system is given in the matrix-notation. The system L(y) = 0 is specified by a matrix L which elements are Maple operators/procedures of an integer argument, for example, k. They compute the coefficients of x^k as a polynomial of θ= x d/dx.

FormalSol.mw - the Maple session file with an example of find formal solutions, when the system is given in the diff-notation. The system L(y) = 0 is specified by a differential equation with power series coefficients:

Sum(a0(k)*x^k, k=0..∞).y(x) + Sum(a1(k)*x^k, k=0..∞).diff(y(x), x) + … + Sum(ar(k)*x^k, k=0..∞).diff(y(x), x\$r) = 0

where r is an integer (the system's order), a0, a1, …, ar are Maple operators/procedures of an integer argument. They compute the matrix coefficients of x^k.

slides.pdf - the slides of the talk Sergei Abramov «The EG-family of algorithms and procedures for solving linear differential and difference higher-order systems» in Functional Equations in LIMoges 2015.

theta_form.mw - the Maple session file with an example of find formal, regular and Laurent solutions, when the system is given in the theta-notation. The system L(y) = 0 is specified by a θ-equation with matrix coefficients:

A0.y(x) + A1.θ(y(x), x) + … + Ar.θ(y(x), x\$r) = 0

where A0, A1, …, Ar are matrices which elements are power series of x. One of possible form of them is Sum(f(k)*x^k, k = 0 .. ∞) where f is a Maple operator/procedure of an integer argument. It computes the coefficients of x^k.

The EG_delta, EG_sigma, Singsys, LaurentSolution, RegularSolution procedures were implemented by D.E.Khmelnov. The FormalSolution procedure was implemented by A.A.Ryabenko.