  Different types of convolution operations involving large Vandermonde
matrices are considered. The convolutions parallel those of large Gaussian
matrices and additive and multiplicative free convolution. First additive and
multiplicative convolution of Vandermonde matrices and deterministic diagonal
matrices are considered. After this, several cases of additive and
multiplicative convolution of two independent Vandermonde matrices are
considered. It is also shown that the convergence of any combination of
Vandermonde matrices is almost sure. We will divide the considered convolutions
into two types: those which depend on the phase distribution of the Vandermonde
matrices, and those which depend only on the spectra of the matrices. A general
criterion is presented to find which type applies for any given convolution. A
simulation is presented, verifying the results. Implementations of all
considered convolutions are provided and discussed, together with the
challenges in making these implementations efficient. The implementation is
based on the technique of Fourier-Motzkin elimination, and is quite general as
it can be applied to virtually any combination of Vandermonde matrices.
Generalizations to related random matrices, such as Toeplitz and Hankel
matrices, are also discussed.
