Asymptotic behavior of a Poincare recurrence system

We establish asymptotic formulae for the solutions of the first order recurrence system x(n+1) = (A + B-n) x(n), where A and B-n (n = 0, 1,...) are square matrices and Sigma(n=o)(infinity) parallel to B-n parallel to(2) < infinity. As a consequence, we confirm a recent conjecture about the asymptotic behavior of the solutions of the higher order scalar equation u(n + 1) = Sigma(i=o)(k) (c(i) + d(i)(n)) u(n - i). (C) 1997 Academic Press.