Oscillation criteria for second-order linear difference equations

A non-trivial solution of (1) is called oscillatory if for every N > 0 there exists an n > N such that X,X n + , 6 0. If one non-trivial solution of (1) is oscillatory then, by virtue of Sturm’s separation theorem for difference equations (see, e.g., [S]), all non-trivial solutions are oscillatory, so, in studying the question of whether a solution {x,> of (1) is oscillatory, it is no restriction to assume the initial conditions