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  "body": "\n## Definition\n\nLet $R(n, x, y)$ be a [[rational function|0XDK9CGYKJ]] of $x$ and $y$ whose coefficients depend on $n \\in \\Z$.\nConsider the second-order difference equation\n$$x_{n+1} = R(n, x_n, x_{n-1}), \\qquad n \\in \\Z.$$\nIt is a *discrete Painlevé equation* if the following hold.\n\n1. the map $(x_{n-1}, x_n) \\mapsto (x_n, x_{n+1})$ is birational for each $n$;\n2. the equation is integrable, for example in the sense that it is the compatibility condition of a Lax pair of linear problems;\n3. the equation has one of the [[Painlevé equations|AZ17JCMRKB]] as a continuum limit.\n\nIn Sakai's geometric classification, these equations arise as translations of affine Weyl groups acting on families of rational surfaces.\nThe dependence on $n$ may also enter through $q^n$ for a fixed $q$, which gives $q$-discrete Painlevé equations.\n\n## Remarks\n\nExample: the equation called d-$\\mathrm{P_I}$,\n$$x_{n+1} + x_n + x_{n-1} = \\frac{\\alpha n + \\beta}{x_n} + \\gamma,$$\nwith constants $\\alpha \\ne 0$, $\\beta$, $\\gamma$, is a discrete Painlevé equation; under a suitable scaling it tends to $\\mathrm{P_I}$, $y'' = 6y^2 + t$.\n\nNon-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.\n\nClause 2 has several versions in the literature, so the definition is less rigid than most.\nA practical test is singularity confinement: a singularity met at one step disappears after finitely many steps.\nAnother standard example is d-$\\mathrm{P_{II}}$, $x_{n+1} + x_{n-1} = \\dfrac{(\\alpha n + \\beta) x_n + a}{1 - x_n^2}$.\n",
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