Sturm comparison theorem of difference equations (Q1907090)

A Sturm comparison theorem for the second order difference equation of this form \[ \Delta^2 U_n + p_n U_{n + 1} + \sum^m_{k = 1} p_n^{(k)} U_{\tau_k (n)} = 0 \] is established, where \(p_n\), \(p_n^{(k)} \in \mathbb{R}\), \(\tau_k (n)\) are integers with \(\lim_{n \to \infty} \tau_k (n) = \infty\), \(k = 1,2, \dots, m\); \(n = N\), \(N + 1, \dots\).