A family of linearisable recurrences with the Laurent property

TL;DR

This paper introduces a family of nonlinear recurrences with the Laurent property, demonstrating their linearisability through two methods, and explores their connections to Laurent phenomenon algebras and the dressing chain for Schrödinger operators.

Contribution

It extends the understanding of Laurent property recurrences by showing their linearisability and linking them to Laurent phenomenon algebras and integrable systems.

Findings

Recurrences possess the Laurent property despite not arising from cluster mutations.

Each recurrence is linearisable via two distinct methods.

Connections to dressing chains for Schrödinger operators are established.

Abstract

We consider a family of nonlinear recurrences with the Laurent property. Although these recurrences are not generated by mutations in a cluster algebra, they fit within the broader framework of Laurent phenomenon algebras, as introduced recently by Lam and Pylyavskyy. Furthermore, each member of this family is shown to be linearisable in two different ways, in the sense that its iterates satisfy both a linear relation with constant coefficients and a linear relation with periodic coefficients. Associated monodromy matrices and first integrals are constructed, and the connection with the dressing chain for Schrodinger operators is also explained.