  The eigenvalue probability density function (PDF) for the Gaussian unitary
ensemble has a well known analogy with the Boltzmann factor for a classical
log-gas with pair potential $- \log | x - y|$, confined by a one-body harmonic
potential. A generalisation is to replace the pair potential by $- \log |\sinh
(\pi (x-y)/L) |$. The resulting PDF first appeared in the statistical physics
literature in relation to non-intersecting Brownian walkers, equally spaced at
time $t=0$, and subsequently in the study of quantum many body systems of the
Calogero-Sutherland type, and also in Chern-Simons field theory. It is an
example of a determinantal point process with correlation kernel based on the
Stieltjes--Wigert polynomials. We take up the problem of determining the
moments of this ensemble, and find an exact expression in terms of a particular
little $q$-Jacobi polynomial. From their large $N$ form, the global density can
be computed. Previous work has evaluated the edge scaling limit of the
correlation kernel in terms of the Ramanujan ($q$-Airy) function. We show how
in a particular $L \to \infty$ scaling limit, this reduces to the Airy kernel.
