  We generally study the density of eigenvalues in unitary ensembles of random
matrices from the recurrence coefficients with regularly varying conditions for
the orthogonal polynomials. First we calculate directly the moments of the
density. Then, by studying some deformation of the moments, we get a family of
differential equations of first order which the densities satisfy (see Theorem
1.2), and give the densities by solving them. Further, we prove that the
density is invariant after the polynomial perturbation of the weight function
(see Theorem 1.5).
