Quantum Painlevé Equations: from Continuous to Discrete - Nagoya, H.; Grammaticos, B.; Ramani, A. (Symmetry, Integrability and Geometry: Methods and Applications, 2008)
We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations.