Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances

We consider the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a distinct given covariance matrix. Such a setup appears frequently in multivariate statistics and enjoys various applications. Only in the degenerate case of two equal covariance matrices H reduces to a single rectangular correlated Wishart matrix, whose spectral statistics is known. Our starting point are recent results by Kumar for the distribution of the matrix H valid in the non-degenerate case. It is given by a confluent hypergeometric function of matrix argument. In the half-degenerate case, when one of the covariance matrices is proportional to the identity, the joint probability density of the eigenvalues of H reduces to a bi-orthogonal ensemble containing ordinary hypergeometric functions. We compute all spectral k-point density correlation functions of H for arbitrary size N. In the half-degenerate case they are given by a determinant of size k of a kernel of certain bi-orthogonal functions. The latter follows from computing the expectation value of a single characteristic polynomial. In the non-degenerate case using superbosonisation techniques we compute the generating function for the k-point resolvent given by the expectation value of ratios of characteristic polynomials.