Reduction of order, periodicity and boundedness in a class of nonlinear, higher order difference equations

Non-autonomous, higher order difference equations of type xn+1=∑i=0kaixn−i+gn(∑i=0kbixn−i) with real variables and parameters have appeared frequently in the literature. We extend some recent results on semiconjugate factorization and reduction of order to cases where characteristic polynomials of the linear expressions ∑i=0kaiui and ∑i=0kbiui have complex roots. This extension yields new results on boundedness and existence of periodic solutions for equations of order 3 or greater.