On a set of convergent solutions for a system of second order linear difference equations

Abstract

The problem was proposed and formulated by the junior author, and the crucial point of the convergence proof was done by the senior author. In (2), A j ( j = Q, 1, 2) are n by n constant matrices and X is an ^-dimensional vector, and t is a complex variable. The system has 2 singular points in the closed complex plane; one at the origin, which is a regular singular point, the other at infinity which is an irregular singular point. We will assume the eigen-values h, Az, -•-, hn of the matrix A^ are mutually distinct, and they satisfy the following pentagonal condition