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We establish a representation of the joint moments of the characteristicpolynomial of a CUE random matrix and its derivative in terms of a solution ofthe sigma-Painleve V equation. The derivation involves the analysis of aformula for the joint moments in terms of a determinant of generalised Laguerrepolynomials using the Riemann-Hilbert method. We use this connection with thesigma-Painleve V equation to derive explicit formulae for the joint moments andto show that in the large-matrix limit the joint moments are related to asolution of the sigma-Painleve III equation. Using the conformal blockexpansion of the tau-functions associated with the sigma-Painleve V and thesigma-Painleve III equations leads to general conjectures for the jointmoments.

A representation of joint moments of CUE characteristic polynomials in terms of Painleve functions