Some Periodic and Non-Periodic Recursions

Abstract.

 It is well known that the recurrence relations

are periodic, in the sense that they define periodic sequences for all choices of the initial data, and lead to sequences with periods 2, 5 and 8, respectively. In this paper we determine all periodic recursions of the form

where are complex numbers, are non-zero and . We find that, apart from the three recursions listed above, only

lead to periodic sequences (with periods 6 and 8). The non-periodicity of (R) when (or and ) depends on the connection between (R) and the recurrence relations

and

We investigate these recursions together with the related

Each of (A), (B), and (C) leads to periodic sequences if k = 1 (with periods 6, 5, and 9, respectively). Also, for k = 2, (B) leads to periodicity with period 8. However, no other cases give rise to periodicity. We also prove that every real sequence satisfying any of (A), (B), and (C) must be bounded. As a consequence, we find that for an arbitrary k, every rational sequence satisfying any of (A), (B), and (C) must be periodic.