  Very recently we have shown that the spherical transform is a convenient tool
for studying the relation between the joint density of the singular values and
that of the eigenvalues for bi-unitarily invariant random matrices. In the
present work we discuss the implications of these results for products of
random matrices. In particular, we derive a transformation formula for the
joint densities of a product of two independent bi-unitarily invariant random
matrices, the first from a polynomial ensemble and the second from a polynomial
ensemble of derivative type. This allows us to re-derive and generalize a
number of recent results in random matrix theory, including a transformation
formula for the kernels of the corresponding determinantal point processes.
Starting from these results, we construct a continuous family of random matrix
ensembles interpolating between the products of different numbers of Ginibre
matrices and inverse Ginibre matrices. Furthermore, we make contact to the
asymptotic distribution of the Lyapunov exponents of the products of a large
number of bi-unitarily invariant random matrices of fixed dimension.
