  We consider ensembles of random matrices, known as biorthogonal ensembles,
whose eigenvalue probability density function can be written as a product of
two determinants. These systems are closely related to multiple orthogonal
functions. It is known that the eigenvalue correlation functions of such
ensembles 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.
