Height growth of solutions and a discrete Painlevé equation

A. Al-Ghassani, R. G. Halburd

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


Consider the discrete equation y n+1 + y n-1 = an + bnyn + cny2n/1-y2n where the right side is of degree two in yn and where the coefficients an, bn and cn are rational functions of n with rational coefficients. Suppose that there is a solution such that for all sufficiently large n, and the height of yn dominates the height of the coefficient functions an, bn and cn. We show that if the logarithmic height of yn grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dPII or its autonomous version or yn is also an admissible solution of a discrete Riccati equation. This provides further evidence that slow height growth is a good detector of integrability.

Original languageEnglish
Pages (from-to)2379-2396
Number of pages18
Issue number7
Publication statusPublished - Jul 1 2015


  • Diophantine integrability
  • discrete Painlevé equations
  • discrete integrable systems

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics


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