  It has been shown recently [10] that Cauchy transforms of orthogonal
polynomials appear naturally in general correlation functions containing ratios
of characteristic polynomials of random NxN Hermitian matrices. Our main goal
is to investigate the issue of universality of large N asymptotics for those
Cauchy transforms for a wide class of weight functions. Our analysis covers
three different scaling regimes: the "hard edge", the "bulk" and the "soft
edge" of the spectrum, thus extending the earlier results known for the bulk.
The principal tool is to show that for finite matrix size N the auxiliary "wave
functions" associated with the Cauchy transforms obey the same second order
differential equation as those associated with the orthogonal polynomials
themselves.
