Basor, EstelleBleher, PavelBuckingham, RobertGrava, TamaraIts, AlexanderIts, ElizabethKeating, Jonathan P.2020-08-282020-08-282019-10Basor, E., Bleher, P., Buckingham, R., Grava, T., Its, A., Its, E., & Keating, J. P. (2019). A representation of joint moments of CUE characteristic polynomials in terms of Painlevé functions. Nonlinearity, 32(10), 4033–4078. https://doi.org/10.1088/1361-6544/ab28c7https://hdl.handle.net/1805/23728We establish a representation of the joint moments of the characteristic polynomial of a CUE random matrix and its derivative in terms of a solution of the -Painlevé V equation. The derivation involves the analysis of a formula for the joint moments in terms of a determinant of generalised Laguerre polynomials using the Riemann–Hilbert method. We use this connection with the -Painlevé V equation to derive explicit formulae for the joint moments and to show that in the large-matrix limit the joint moments are related to a solution of the -Painlevé III equation. Using the conformal block expansion of the -functions associated with the -Painlevé V and the -Painlevé III equations leads to general conjectures for the joint moments.enPublisher Policyjoint momentsCUE random matrixPainlevé functionsArticle