Rational Solutions of Difference Painlevé Equations

Abstract

We capture all the rational solutions of some difference Painlevé equations of P$_{\mathrm{I}}$ and P$_{\mathrm{II}}$ types. For non-autonomous cases, it is shown that all the rational solutions of the difference P$_{\mathrm{II}}$ are ones generated by successive application of auto-Bäcklund transformations to the seed solution vanishing identically, and that the other equations of P$_{\mathrm{I}}$ type admit no rational solutions. For autonomous cases, all the nontrivial rational solutions are obtained, and they exist under a certain condition on a fixed point of the equation. If such a condition is not satisfied, there exist solutions that are rational in an exponential function.