Abstract
We give a new representation of solutions of the periodic linear difference equation of the form x(n + 1) = Bx(n) + b(n), where B is a complex p × p matrix and b(n) ∈ Cp satisfies the condition b(n) = b(n + ρ), ρ ∈ N, ρ ≥ 2. If B = eτA, τ > 0, then the equation has two representations of solutions based on A and B. In particular, the representation of solutions based on A is deduced from the one based on B by using the translation formulae from B to A. Using these representations, we can obtain the complete classification of the set of initial values according to the behavior of solutions. As applications of these results, by the initial values we characterize necessary and sufficient conditions on the existence of a bounded solution and a ρ-periodic solution.
