A note on biorthogonal ensembles

We study multiple orthogonal polynomials in the context of biorthogonal ensembles of random matrices. In these ensembles, the eigenvalue probability density function factorizes into a product of two determinants while the eigenvalue correlation functions can be written as a determinant of a kernel function. We show that the kernel is itself an average of a single ratio of characteristic polynomials. In the same vein, we prove that the type I multiple polynomials can be expressed as an average of the inverse of a characteristic polynomial. We finally introduce a new biorthogonal matrix ensemble, namely the chiral unitary perturbed by a source term, whose multiple polynomials are related to the modified Bessel function of the first kind.