Auto-Bäcklund transformations and special integrals for differential-delay Painlevé hierarchies

•Special integrals for differential-delay Painlevé hierarchies.•Auto-Bäcklund transformation for a differential-delay Painlevé hierarchy.•Integrability properties of lattice hierarchies used to derive our results.•Results also applicable to non-integrable cases.

The six Painlevé equations have attracted much interest over the last thirty years or so. More recently many authors have begun to explore properties of higher-order versions of both these equations and their discrete analogues. However, little attention has been paid to differential-delay Painlevé equations, i.e., analogues of the Painlevé equations involving both shifts in and derivatives with respect to the independent variable, and even less to higher-order analogues of these last. In the current paper we discuss the phenomenon whereby members of one differential-delay Painlevé hierarchy define solutions of higher-order members of a second differential-delay Painlevé hierarchy. We also give an auto-Bäcklund transformation for a differential-delay Painlevé hierarchy. The key to our approach is the underlying Hamiltonian structure of related completely integrable lattice hierarchies.