Asymptotic distribution of the spectrum of some symmetric polynomials of unitary invariant random matrix ensembles

We study asymptotic behavior of the spectrum of some symmetric polynomials in unitary invariant random matrix ensembles in the limit of infinite matrix order. We express the limiting normalized eigenvalue counting measures of polynomials in terms of corresponding measures of operands, by deriving functional equations on the proper analytic (Stieltjes) transforms of the measures and compare these equations with the corresponding ones derived before for another ensembles.