Poincaré types solutions of systems of difference equations

Abstract

The study of asymptotics theory of ordinary difference equations originates from the work of Henry Poincar´e. In 1885, Poincar´e [19] published a seminal paper on the asymptotics of both ordinary difference and differential equations, where he studied the k th order linear nonautonomous difference equation of the form y(n + k) + (a1 + p1(n))y(n + k − 1) + · · · + (ak + pk(n))y(n) = 0 (1.1) with k ∈ Z+, ai ∈ C and pi(n) : Z+ → C for 1 ≤ i ≤ k. This equation is said to be of Poincar´e type if limn→∞ pi(n) = 0 for 1 ≤ i ≤ k. We assume that Eq.(1.1) is of Poincar´e type and associated with Eq.(1.1) its limiting equation x(n + k) + a1x(n + k − 1) + · · · + akx(n) = 0 (1.2) with the corresponding characteristic equation λ k + a1λ k−1 + · · · + ak = 0. (1.3) Suppose that λ1, λ2, ..., λk are the characteristic roots of Eq.(1.2), i.e., the roots of Eq.(1.3). It is straightforward to see that solutions of Eq.(1.2) are of the form Xr i=1 qi(n)λ n i