  The $\tau$-function theory of Painlev\'e systems is used to derive
recurrences in the rank $n$ of certain random matrix averages over U(n). These
recurrences involve auxilary quantities which satisfy discrete Painlev\'e
equations. The random matrix averages include cases which can be interpreted as
eigenvalue distributions at the hard edge and in the bulk of matrix ensembles
with unitary symmetry. The recurrences are illustrated by computing the value
of a sequence of these distributions as $n$ varies, and demonstrating
convergence to the value of the appropriate limiting distribution.
