It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensembledM of n×n Hermitian matrices, these kernels have universal behavior at the origin of the spectrum, as n → ∞, in terms of Bessel functions. Our approach is based on the characterization of orthogonal polynomials together with their Cauchy transforms via a matrix Riemann-Hilbert problem, due to Fokas, Its and Kitaev, and on an application of the Deift/Zhou steepest descent method for matrix Riemann-Hilbert problems to obtain the asymptotic behavior of the Riemann-Hilbert problem.