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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

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