# Linear Fractional Recurrences: Periodicities and Integrability

## TL;DR

This paper studies k-step linear fractional recurrences, revealing periodic behaviors and integrability properties, including the specific case of the Lyness process which exhibits quadratic degree growth and known integrals.

## Contribution

It establishes the existence of periodic k-step linear fractional recurrences and analyzes the integrability and degree growth of the Lyness process.

## Key findings

- Existence of k-step recurrences with period 4k
- Lyness process has quadratic degree growth
- Lyness process is integrable with known integrals

## Abstract

We consider k-step recurrences of the form $z_{n+k} = A(z)/B(z)$, where A and B are linear functions of $z_n, z_{n+1}, ..., z_{n+k-1}$, which we call k-step linear fractional recurrences. The first Theorem in this paper shows that for each k there are k-step linear fractional recurrences which are periodic of period 4k. Among this class of recurrences, there is also the so-called Lyness process, which has the form $A(z)/B(z) = (a +z_{n+1} + z_{n+2} + ... + z_{n+k-1})/z_n$. The second Theorem shows that the Lyness process has quadratic degree growth. The Lyness process is integrable, and we discuss its known integrals.