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  "body": "\n## Summary\n\nDiscrete Painlevé equations are integrable nonlinear second-order difference equations that play the role of the [[Painlevé equations|AZ17JCMRKB]] for recurrences.\nThey are non-autonomous, and in a suitable continuum limit they tend to one of the six Painlevé differential equations.\nTheir integrability shows up as singularity confinement, as a Lax pair of linear problems, and as a description by birational maps of rational surfaces.\nThey arose in the theory of orthogonal polynomials and random matrix models, where recurrence coefficients satisfy them.\n\n## Explanations\n\n### L1 human\n\nThe Painlevé equations are special nonlinear differential equations whose solutions behave unusually well.\nDiscrete Painlevé equations are recurrences with the same good behaviour: each new term is a rational function of the two previous ones and of the step number.\nA typical one is $x_{n+1} + x_n + x_{n-1} = (\\alpha n + \\beta)/x_n + \\gamma$, with constants $\\alpha, \\beta, \\gamma$.\nIf a term happens to be $0$, the next terms blow up for a moment, but the recursion recovers after a few steps instead of losing information.\n",
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