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”

Characterization of regular singular linear systems of difference equations

We apply a general process to obtain "logarithms" for some families of natural characters for q-difference or difference operators

Une remarque sur les "logarithmes" associés à certains caractères

We study Turrittin’s formal reduction of systems of linear difference equations of the forms
 
$$Y(x + 1) = x^c A(x)Y(x) or Y(x + 1) - Y(x) = x^{ - r/s} A(x)Y(x)$$

 . Herec is a rational number;r ands are two positive integers; andA(x)=A
 0+A1x−1/s+...∈ gl(n, C[[x
 −1/s]]) withA
 0≠0; in particular, detA(x)≠0 in the first system. For the reduction in the nilpotent case we use Arnold’s normal form of matrices depending on parameters. We present an algorithm to construct such a normal form for difference systems. We will prove that for a system in the second form, the exponential part in a formal fundamental matrix solution depends only on the matrix coefficientsA
 0, ...,A
 n(s-r)−1. Let det(A(x))=x
 −m/s(a0+a1x−1/s+...), a0≠0 for the first system. It is also proved that the Γ part in a formal fundamental matrix solution of the first system depends only on the matrix coefficientsA
 0, ...,A
 m, and generally the irregular part in a formal fundamental matrix solution of the first system depends only on the matrix coefficientsA
 0, ...,A
 m+ns−1. The results are important from an algorithmic point of view. The bounds are best possible.

Forme normale d’Arnold et réduction formelle des systèmes d’équations linéaires aux différences

William Ashton Harris Jr. was born December 18, 1930 in New Orleans, Louisiana, USA. He studied mathematics with minors in physics and appl ied mechanics (elasticity) at the University of Minnesota receiving his Ph.D. in mathematics in 1955 with the thesis "A boundary value problem for a system of ordinary linear differential equations involving powers of a parameter". His thesis advisor was Professor H.L. Turrittin.He remained at the University of Minnesota and became professor in 1968. In 1970 he moved to the University of Southern California where he stayed until his untimely death on January 8, 1998. He has held visiting appointments at several other universities.

The contributions of W.A. Harris Jr, to difference equations

In this paper, we provide a new approach for computing regular solutions of first-order linear difference systems. We use the setting of factorial series known to be very well suited for dealing with difference equations and we introduce a sequence of functions which play the same role as the powers of the logarithm in the differential case. This allows us to adapt the approach of [5] where we have developed an algorithm for computing regular solutions of linear differential systems.

/pdf/a-new-approach-for-computing-regular-solutions-of-linear-18gh9ji580.pdf

A New Approach for Computing Regular Solutions of Linear Difference Systems

Analytic theory of difference equations

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Analytic theory of difference equations