Correlation kernels for sums and products of random matrices

Let X be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of GX and TX, where G is a complex Ginibre matrix and T is a truncated unitary matrix. We also consider the product of X and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum H + M where H is a GUE matrix and M is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of H + M follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.