  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.
