Ruiz Gordoa, Maria del PilarPickering, Andrew2025-03-172025-03-172024P. R. Gordoa, A. Pickering, Higher order nonlinear special functions: Painlevé hierarchies, a survey, in "Progress in Special Functions", Contemporary Mathematics, vol. 807 (Amer. Math. Soc., Providence, RI, 2024), pp. 131-170.9781470474294 (print)978-1-4704-7722-6 (online)https://hdl.handle.net/10115/80057The authors were supported by project PID2020-115273GB-I00 funded by MCIN/AEI/10.13039/501100011033 and grant RED2022-134301-T funded by MCIN/AEI/10.13039/501100011033.The six Painlevé transcendents are widely accepted as nonlinear special functions. Over the last quarter of a century or so, there has been a surge of interest in higher-order analogues of the Painlevé equations, most often defined as members of hierarchies of equations of increasing order, i.e., of so-called Painlevé hierarchies. We give here a survey of such Painlevé hierarchies, including of their derivation and the derivation of their properties. Amongst other aspects, we discuss the relationships between the properties of completely integrable hierarchies, e.g., Hamiltonian structures and Miura maps, nonisospectral scattering problems, and those of Painlevé hierarchies, e.g., Lax pairs, B¨acklund and auto-B¨acklund transformations, and sequences of special solutions. Given the large number of papers published on Painlevé hierarchies, we hope this review will serve as a useful future reference.en-USAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Painlevé equations and hierarchiesBäcklund and auto-Bäcklund transformationscompletely integrable systemsHamiltonian structuresLax pairsBook chapterhttps://doi.org/10.1090/conm/807info:eu-repo/semantics/openAccess