On the Laxton Group

TL;DR

This paper redefines Laxton's group of linear recurrence sequences, clarifies its structure, and provides a natural algebraic number theory interpretation, simplifying proofs and enabling further analysis of these sequences.

Contribution

It introduces a natural redefinition of Laxton's group and determines its structure, extending previous work on quotient groups to the entire group.

Findings

Complete structure of Laxton's group determined

Simplified proofs using algebraic number theory

New interpretation of Laxton's results

Abstract

We redefine a multiplicative group structure on the set of equivalence classes of rational sequences satisfying a fixed linear recurrence of degree two, which was defined by R. R. Laxton in his paper "On groups of linear recurrences I" published in Duke Math. 36, 721--736 (1969). In the article, he also defined some natural subgroups of the group, and determined the structures of their quotient groups. However, he did not study the whole group itself. Nothing has been known about the structure of Laxton's whole group and its interpretation. The aims of this paper are to redefine Laxton's group in a natural way and determine the structure of the whole group itself, which clarifies Laxton's results on the quotient groups. According to our formulation by algebraic number theory method, we can simplify the proof of Laxton's results. Our definition also gives a natural interpretation of Laxton's results, and makes us possible to use the group to show various properties of such sequences.