Anna A. Ryabenko

Procedures for searching local solutions of linear differential systems with infinite power series in the role of coefficients

Construction of Laurent, regular, and formal (exponential–logarithmic) solutions of full-rank linear ordinary differential systems is discussed. The systems may have an arbitrary order, and their coefficients are formal power series given algorithmically. It has been established earlier that the first two problems are algorithmically decidable and the third problem is not decidable. A restricted variant of the third problem was suggested for which the desired algorithm exists. In the paper, a brief survey of algorithms for the abovementioned decidable problems is given. Implementations of these algorithms in the form of Maple procedures with a uniform interface and data representation are suggested.

On exponential-logarithmic solutions of linear differential systems with power series coefficients

The paper studies construction of formal exponential-logarithmic solutions for a system of ordinary linear differential equations the coefficients of which are algorithmically defined formal power series. The construction of all such solutions is generally algorithmically undecidable problem. We propose an algorithm and its implementation in Maple that make it possible to construct a basis of the space of solutions of the full-rank system if the dimension of this space is known.

Resolving Sequence of Operators for Linear Ordinary Differential and Difference Systems of Arbitrary Order

We introduce the notion of a resolving sequence of (scalar) operators for a given differential or difference system with coefficients in some differential or difference field K. We propose an algorithm to construct, such a sequence, and give some examples of the use of this sequence as a suitable auxiliary tool for finding certain kinds of solutions of differential and difference systems of arbitrary order. Some experiments with our implementation of the algorithm are reported.

Hypergeometric Solutions of First-Order Linear Difference Systems with Rational-Function Coefficients

Algorithms for finding hypergeometric solutions of scalar linear difference equations with rational-function coefficients are known in computer algebra. We propose an algorithm for the case of a first-order system of such equations. The algorithm is based on the resolving procedure which is proposed as a suitable auxiliary tool, and on the search for hypergeometric solutions of scalar equations as well as on the search for rational solutions of systems with rational-function coefficients. We report some experiments with our implementation of the algorithm.

A definite summation of hypergeometric terms of special kind

A definite summation of hypergeometric terms of two variables is considered. Currently, for summation of such terms in computer algebra systems, a combination of the Zeilberger algorithm and the discrete Newton-Leibniz formula is used. As is known, the result of such summation is not always correct. In the paper, it is proposed to use simple preliminary examination of the term before applying the Zeilberger algorithm. If the hypergeometric term belongs to the class described in the paper, then the result obtained by the Zeilberger algorithm is correct or even there is no need to use it at all.

Symbolic solution of nonhomogeneous linear ordinary differential equations in terms of power series

AbstractFor a nonhomogeneous linear ordinary differential equation Ly(x) = f(x) with polynomial coefficients and a holonomic right-hand side, a set of points x = a is found where a power series solution

Revealing matrices of linear differential systems of arbitrary order

y(x) = \sum\nolimits_{n = 0}^\infty {c_n (x - a)} ^n

Hypergeometric pattern matching summation in maple

with hypergeometric coefficients exists (starting from some number, the ratio cn + 1/cn is a rational function of n).