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

We 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 -Painlev´e 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 -Painlev´e 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 -Painlev´e III0 equation. Using the conformal block expansion of the ⌧ -functions associated with the -Painlev´e V and the -Painlev´e III0 equations leads to general conjectures for the joint moments.