  We investigate a one-parameter family of Coulomb gases in two dimensions,
which are confined to an ellipse, due to a hard wall constraint, and are
subject to an additional external potential. At inverse temperature $\beta=2$
we can use the technique of planar orthogonal polynomials, borrowed from random
matrix theory, to explicitly determine all $k$-point correlation functions for
a fixed number of particles $N$. These are given by the determinant of the
kernel of the corresponding orthogonal polynomials, which in our case are the
Gegenbauer polynomials, or a subset of the asymmetric Jacobi polynomials,
depending on the choice of external potential, as shown in a companion paper
recently published by three of the authors. In the rotationally invariant case,
when the ellipse becomes the unit disc, our findings agree with that of the
ensemble of truncated unitary random matrices. The thermodynamical large-$N$
limit is investigated in the local scaling regime in the bulk and at the edge
of the spectrum at weak and strong non-Hermiticity. We find new universality
classes in these limits and recover the sine- and Bessel-kernel in the
Hermitian limit. The limiting global correlation functions of particles in the
interior of the ellipse are more difficult to obtain but found in the special
cases corresponding to the Chebyshev polynomials.
