  We study the densities of limiting distributions of squared singular values
of high-dimensional matrix products composed of independent complex Gaussian
(complex Ginibre) and truncated unitary matrices which are taken from Haar
distributed unitary matrices with appropriate dimensional growth. In the
general case we develop a new approach to obtain complex integral
representations for densities of measures whose Stieltjes transforms satisfy
algebraic equations of a certain type. In the special cases in which at most
one factor of the product is a complex Gaussian we derive elementary
expressions for the limiting densities using suitable parameterizations for the
spectral variable. Moreover, in all cases we study the behavior of the
densities at the boundary of the spectrum.
