Some periodic and non-periodic recursions, Marianna Csornyei (UCL)

It is well known that the recurrence relations

x_{n} = \frac{1}{x_{n−1}} , x_{n} = \frac{1 + x_{n−1}}{x_{n−2}} and x_{n} = \frac{1 + x_{n−1} + x_{n−2}}{x_{n−3}}

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. We determine all periodic recursions of the form

x_{n} = \frac{a_{0} + a_{1}x_{n−1} + ... + a_{k}x_{n−k}}{x_{n−k−1}} ,  (R)

where a_{0}, ..., a_{k} are complex numbers, a_{1}, ..., a_{k−1} are non-zero and a_{k} = 1.

The non-periodicity of (R) depends on the connection between (R) and the recurrence relations

x_{n} = max(x_{n−1}, ..., x_{n−k}) − x_{n−k−1}  (A)

and

x_{n} = max(x_{n−1}, ..., x_{n−k}, 0) − x_{n−k−1}.  (B)

We investigate these recursions together with the related

x_{n} = | max(x_{n−1}, ..., x_{n−k})| − x_{n−k−1}.  (C)

Each of (A), (B), and (C) leads to periodic sequences if k = 1. Also, for k = 2, (B) leads to a periodic sequence. However, we prove that no other cases give rise to periodicity.