### Abstract

Consider the discrete equation y _{n+1} + y _{n-1} = a_{n} + b_{n}y_{n} + c_{n}y^{2}_{n}/1-y^{2}_{n} where the right side is of degree two in y_{n} and where the coefficients a_{n}, b_{n} and c_{n} 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 y_{n} dominates the height of the coefficient functions a_{n}, b_{n} and c_{n}. We show that if the logarithmic height of y_{n} grows no faster than a power of n then either the equation is a well known discrete Painlevé equation dP_{II} or its autonomous version or y_{n} 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 language | English |
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Pages (from-to) | 2379-2396 |

Number of pages | 18 |

Journal | Nonlinearity |

Volume | 28 |

Issue number | 7 |

DOIs | |

Publication status | Published - Jul 1 2015 |

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### Keywords

- Diophantine integrability
- discrete integrable systems
- discrete Painlevé equations

### ASJC Scopus subject areas

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

### Cite this

*Nonlinearity*,

*28*(7), 2379-2396. https://doi.org/10.1088/0951-7715/28/7/2379