An exact formula for general spectral correlation function of random Hermitian matrices

We have found an exact formula for a general correlation function containing both products and ratios of integer powers of characteristic polynomials of random Hermitian matrices. The answer is given in the form of a (small size) determinant and is valid for any invariant ensemble of $\beta=2$ symmetry class. Essential difference from the previously studied correlation functions (of products only) is the appearance of non-polynomial functions along with the orthogonal polynomials. These non-polynomial functions are the Cauchy transforms of the orthogonal polynomials.