Height growth of solutions and a discrete Painlev\'e equation

Abstract

Consider the discrete equation $$ y_{n+1}+y_{n-1}=\frac{a_n+b_ny_n+c_ny_n^2}{1-y_n^2}, $$ 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 $n$, $y_n\in\mathbb{Q}$ 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 Painlev\'e equation ${\rm dP}_{\!\rm II}$ 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.