Discrete Painlevé equations

Remarks

Example: the equation called d-$\mathrm{P_I}$, $$x_{n+1} + x_n + x_{n-1} = \frac{\alpha n + \beta}{x_n} + \gamma,$$ with constants $\alpha \ne 0$, $\beta$, $\gamma$, is a discrete Painlevé equation; under a suitable scaling it tends to $\mathrm{P_I}$, $y'' = 6y^2 + t$.

Non-example: the logistic recurrence $x_{n+1} = r x_n (1 - x_n)$ is a rational recurrence, but its map $x \mapsto r x (1 - x)$ is two-to-one, so it is not birational and clause 1 fails.

Clause 2 has several versions in the literature, so the definition is less rigid than most. A practical test is singularity confinement: a singularity met at one step disappears after finitely many steps. Another standard example is d-$\mathrm{P_{II}}$, $x_{n+1} + x_{n-1} = \dfrac{(\alpha n + \beta) x_n + a}{1 - x_n^2}$.