Periodicity of some classes of holomorphic difference equations

In this paper, we investigate the periodicity character of some classes of difference equations. Among other results, we give a simpler proof of an Open Problem addressed in Csornyei and Laczkovich, 2001, "Some periodic and non-periodic recursions", Monath. Math., 132, 215-236. We also consider Open Problem 2.8 in Grove and Ladas, 2005, Periodicities in Nonlinear Difference Equations, (Chapman and Hall/CRC), and the case when a difference equation has periodic solutions depending on arbitrary parameters.