  We show that the ratio of a discrete Toeplitz/Hankel determinant and its
continuous counterpart equals a Freholm determinant involving continuous
orthogonal polynomials. This identity is used to evaluate a triple asymptotic
of some discrete Toeplitz/Hankel determinants which arise in studying
non-intersecting processes. We show that the asymptotic fluctuations of the
width of such processes are given by the GUE Tracy-Widom distribution. This
result leads us to an identity between the GUE Tracy-Widom distribution and the
maximum of the sum of two independent Airy processes minus a parabola. We
provide an independent proof of this identity.
