From discrete integrable equations to Laurent recurrences

Abstract

We show how to obtain relations for the divisors of terms generated by a homogenized version of a rational recurrence. When the rational recurrence confines singularities the relations take the form of a rational recurrence, possibly with periodic coefficients. As the recurrence generates polynomials one expects it to possess the Laurent property. The method we develop uses ultra-discretization and recursive factorization. It is applied to certain QRT-maps which gives rise to Somos-k ($k=4,5$) sequences with periodic coefficients. Novel $(N+3)$-rd order recurrences are obtained from the Nth order DTKQ-equation ($N=2,3$). In each case the resulting recurrence equation has the Laurent property. The method is equally applicable to non-integrable or non-confining equations. However, in the latter case the degree and the order of the relation might display unbounded growth. We demonstrate the difference, by considering different parameter choices in a generalized Lyness equation.

Keywords:

• Difference equations
• integrability
• Laurent recurrences
• singularity confinement
• ultra-discretization
• QRT-map
• Somos sequence
• generalized Lyness equation

Acknowledgements

Both authors acknowledge useful discussions with Reinout Quispel. We thank Ralph Willox for bringing to our attention reference [40], and thank the referees for some useful remarks and additional references.

Notes

No potential conflict of interest was reported by the authors.

1 Notation: a periodic function $p_{n+m}=p_n$ is defined by m values: with $p_{\text{mod}m}=[v_1,\dots ,v_m]$ we mean $p_n=v_{n\text{mod}m}$.

Additional information

Funding

This research was supported by the Australian Research Council and by the La Trobe University Disciplinary Research Program in Mathematical and Computer Sciences.