Linear Perturbations of a Nonoscillatory Second Order Difference Equation

Let x1n and x2n be recessive and dominant solutions of the nonoscillatory difference equation rn−1 xn−1 + pnxn = 0.It is shown that if ∞ fnx1nx2n converges (perhaps conditionally) and satisfies a second condition on its order of covergence, then the difference equation rn−1 yn−1 +pnyn = fnyn is also nonoscillatory and has recessive and dominant solutions y1n and y2n that behave asymptotically like x1n and x2n