  We consider a class of N=2 conformal SU(N) SYM theories in four dimensions
with matter in the fundamental, two-index symmetric and anti-symmetric
representations, and study the corresponding matrix model provided by
localization on a sphere S4, which also encodes information on flat-space
observables involving chiral operators and circular BPS Wilson loops. We review
and improve known techniques for studying the matrix model in the large-N
limit, deriving explicit expressions in perturbation theory for these
observables. We exploit both recursive methods in the so-called full Lie
algebra approach and the more standard Cartan sub-algebra approach based on the
eigenvalue distribution. The sub-class of conformal theories for which the
number of fundamental hypermultiplets does not scale with N differs in the
planar limit from the N=4 SYM theory only in observables involving chiral
operators of odd dimension. In this case we are able to derive compact
expressions which allow to push the small 't Hooft coupling expansion to very
high orders. We argue that the perturbative series have a finite radius of
convergence and extrapolate them numerically to intermediate couplings. This is
preliminary to an analytic investigation of the strong coupling behavior, which
would be very interesting given that for such theories holographic duals have
been proposed.
