  Statistics over the Gaussian unitary ensemble and the Wishart ensemble of
random matrices often have nice closed-form expressions. These are related to
multivariate extensions of the Hermite, Laguerre, and Jacobi polynomials, which
often occur in the study of these ensembles.
  In the paper, we develop a formal $q$-analog of the Gaussian unitary
ensemble, using $q$-Hermite polynomials and coefficient extraction instead of
integration. This way we derive $q$-analogs for many well-known eigenvalue
statistics. One of these is related to the Harer-Zagier formula, which uses a
matrix integral to count the number of unicellular maps on $n$ vertices by
genus.
