Asymptotic summation for second-order finite difference systems

Conditional summable hypotheses are given to obtain asymptotic summation of some 2 × 2 difference systems. We obtain asymptotic formulae of the solutions of a perturbed system, knowing the recessive and dominant solutions of the unperturbed system. The Casoratian is generally nonconstant and nonconvergent and it appears in the asymptotic formulas