One way in which difference Painlevé equations arise is in the study of difference equations admitting meromorphic solutions of slow growth in the sense of Nevanlinna theory. The idea that the existence of sufficiently many finite-order meromorphic solutions could be considered as a version of the Painlevé property for difference equations was introduced by Ablowitz, Halburd and Herbst. In this talk necessary conditions are obtained for certain types of delay differential equations to admit a transcendental meromorphic solution of hyper-order less than one. The equations obtained include delay Painlevé equations and equations solvable by elliptic functions. We conclude with recent results on the existence of transcendental meromorphic solutions of first-order difference equations, without growth conditions.

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