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On the General Solution of the Heideman–Hogan Family of Recurrences

Part of: Difference equations Difference and functional equations Arithmetic rings and other special rings Sequences and sets

Published online by Cambridge University Press:  14 August 2018

Andrew N. W. Hone  and
Chloe Ward
Andrew N. W. Hone
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK (a.n.w.hone@kent.ac.uk)
Chloe Ward
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK (a.n.w.hone@kent.ac.uk)
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Abstract

We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.

Keywords

nonlinear recurrenceLaurent propertylinearization

MSC classification

Primary: 39A20: Multiplicative and other generalized difference equations, e.g. of Lyness type
Secondary: 11B37: Recurrences 13F60: Cluster algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 4 , November 2018 , pp. 1113 - 1125
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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