Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-26T04:32:09.818Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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 ([email protected])
Chloe Ward
Affiliation:
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, UK ([email protected])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Alman, J., Cuenca, C. and Huang, J., Laurent phenomenon sequences, J. Algebr. Comb. 43 (2016), 589–633.Google Scholar
2Assem, I., Reutenauer, C. and Smith, D., Friezes, Adv. Math. 225 (2010), 3134–3165.Google Scholar
3Byrnes, G. B., Haggar, F. and Quispel, G. R. W., Sufficient conditions for dynamical systems to have pre-symplectic or pre-implectic structures, Physica A 272 (1999), 99–129.Google Scholar
4Di Francesco, P. and Kedem, R., Q-systems, heaps, paths and cluster positivity, Comm. Math. Phys. 293 (2010), 727–802.Google Scholar
5Dodgson, C. L., Condensation of determinants, Proc. R. Soc. Lond. 15 (1866), 150–155.Google Scholar
6Everest, G., van der Poorten, A., Shparlinski, I. and Ward, T., Recurrence sequences, AMS Mathematical Surveys and Monographs, Volume 104 (American Mathematical Society, Providence, RI, 2003).Google Scholar
7Fomin, S. and Zelevinsky, A., The Laurent phenomenon, Adv. Appl. Math. 28 (2002), 119–144.Google Scholar
8Fordy, A. P., Mutation-periodic quivers, integrable maps and associated Poisson algebras, Philos. Trans. R. Soc. Lond. Ser. A 369 (2010), 1264–1279.Google Scholar
9Fordy, A. P. and Marsh, R. J., Cluster mutation-periodic quivers and associated Laurent sequences, J. Algebr. Comb. 34 (2009), 19–66.Google Scholar
10Fordy, A. P. and Hone, A. N. W., Discrete integrable systems and Poisson algebras from cluster maps, Commun. Math. Phys. 325 (2014), 527–584.Google Scholar
11Gale, D., The strange and surprising saga of the Somos sequences, Math. Intelligencer 13(1) (1991), 40–42; Somos sequence update, Math. Intelligencer 13(4) (1991), 49–50.Google Scholar
12Heideman, P. and Hogan, E., A new family of Somos-like recurrences, Electron. J. Comb. 15 (2008), #R54, 8pp.Google Scholar
13Hogan, E., Experimental mathematics applied to the study of non-linear recurrences, PhD thesis, Rutgers University, 2011.Google Scholar
14Hone, A. N. W., Nonlinear recurrence sequences and Laurent polynomials, in Number theory and polynomials (eds McKee, J. and Smyth, C.), LMS Lecture Note Series, Volume 352, pp. 188–210 (Cambridge University Press, Cambridge, 2008).Google Scholar
15Hone, A. N. W. and Ward, C., A family of linearizable recurrences with the Laurent property, Bull. Lond. Math. Soc. 46 (2014), 503–516.Google Scholar
16Keller, B. and Scherotzke, S., Linear recurrence relations for cluster variables of affine quivers, Adv. Math. 228 (2011), 1842–1862.Google Scholar
17Lam, T. and Pylyavskyy, P., Laurent phenomenon algebras, Camb. J. Math. 4 (2016), 121–162.Google Scholar
18Lamb, J. S. W. and Roberts, J. A. G., Time-reversal symmetry in dynamical systems: a survey, Physica D 112 (1998), 1–39.Google Scholar
19Ward, C., Discrete integrability and nonlinear recurrences with the Laurent property, PhD thesis, University of Kent, 2013.Google Scholar