Nonoscillatory solutions of a second-order difference equation of Poincaré type (Q1021797)

For the difference equation \[ x_{n+2}+b_nx_{n+1}+c_nx_n=0 \] with real coefficients satisfying \(b_n\to\beta<0\), \(c_n\to\beta^2/4\) as \(n \to\infty\), it is shown that every non-oscillatory solution has the Poincaré property \(\frac{x_{n+1}}{x_n}\to\beta\). Note that \(\beta\) is a double zero of the corresponding characteristic equation.