  We study the local properties of eigenvalues for the Hermite (Gaussian),
Laguerre (Chiral) and Jacobi $\beta$-ensembles of $N\times N$ random matrices.
More specifically, we calculate scaling limits of the expectation value of
products of characteristic polynomials as $N\to\infty$. In the bulk of the
spectrum of each $\beta$-ensemble, the same scaling limit is found to be
$e^{p_{1}}{}_1F_{1}$ whose exact expansion in terms of Jack polynomials is well
known. The scaling limit at the soft edge of the spectrum for the Hermite and
Laguerre $\beta$-ensembles is shown to be a multivariate Airy function, which
is defined as a generalized Kontsevich integral. As corollaries, when $\beta$
is even, scaling limits of the $k$-point correlation functions for the three
ensembles are obtained. The asymptotics of the multivariate Airy function for
large and small arguments is also given. All the asymptotic results rely on a
generalization of Watson's lemma and the steepest descent method for integrals
of Selberg type.
