Characterization of regular singular linear systems of difference equations

Abstract

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations atz=∞. We are concerned with the behavior of solutions near the pointz=∞ (the only “fixed” singularity for difference equations). It is important to know whether a system of linear difference equations has a regular singularity or an irregular singularity. To a given system (Δ) we can assign a number μ, called the Moser's invariant of (Δ), so that the system is regular singular if and only if μ≤1. We shall develop an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system of difference equations to an “irreducible form”. The computation ot the number μ can be done explicitly from this “irreducible form”.