  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 sigma-Painleve 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
sigma-Painleve 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 sigma-Painleve III equation. Using the conformal block
expansion of the tau-functions associated with the sigma-Painleve V and the
sigma-Painleve III equations leads to general conjectures for the joint
moments.
