2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/69906We obtain uniform asymptotics for polynomials orthogonal on a fixed and varying arc of the unit circle with a positive analytic weight function. We also complete the proof of the large $s$ asymptotic expansion for the Fredholm determinant with the kernel $\sin z/(πz)$ on the interval $[0,s]$, verifying a conjecture of Dyson for the constant term in the expansion. In the Gaussian Unitary Ensemble of random matrices, this determinant describes the probability for an interval of length $s$ in the bulk scaling limit to be free from the eigenvalues.20 pages, 1 figure, small changesFunctional AnalysisMathematical Physicstext