Linear third-order difference equations: oscillatory and asymptotic behavior (Q1802715)

A point of contact of the graph of \(U = \{U_ n\}\) satisfying (1) \(\Delta^ 3U_ n + P_{n+1}\Delta U_{n+2} + Q_ nU_{n+2} = 0\), with the real axis is a node. A solution of (1) is said to be oscillatory if it has arbitrarily large nodes. It is proved that (1) always has an oscillatory solution. A sufficient condition is also given in terms of \(P_ n\) and \(Q_ n\) so that (1) has a solution satisfying, \(\text{Sgn }U_ n = \text{Sgn }\Delta U_ n = \text{Sgn }\Delta^ 2U_ n\).