The discrete first, second and thirty-fourth Painlevé hierarchies

Abstract

The discrete first and second Painlevé equations (dP and dP) are integrable difference equations which have classical first, second or third Painlevé equations (P, P or P) as continuum limits. dP and dP are believed to be integrable because they are discrete isomonodromy conditions for associated (single-valued) linear problems. An infinite hierarchy of integrable difference equations that share the same linear deformation problem as dP was shown to exist by Cresswell and Joshi. In this paper, we recall the results shown for dP and show how to deduce a hierarchy for dP. Each member of the respective hierarchies is shown to be generated by difference recursion operators. Furthermore, we show that continuum limits of these difference hierarchies lead to the P, P and P hierarchies. Finally, we construct Miura transformations of the dP hierarchy and show that these lead to the hierarchy of the discrete thirty-fourth Painlevé equation.