  We derive expansions of the Hermite and Laguerre kernels at the edge of the
spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n
Laguerre Unitary Ensem- ble (LUEn), respectively. Using these large n kernel
expansions, we prove an Edgeworth type theorem for the largest eigenvalue
distribution function of GUEn and LUEn. In our Edgeworth expansion, the
correction terms are expressed in terms of the same Painleve II function
appearing in the leading term, i.e. in the Tracy-Widom distribution. We
conclude with a brief discussion of the universality of these results.
