Edgeworth expansion of the largest eigenvalue distribution function of GUE and LUE

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 ensemble (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 Painlevé II function appearing in the leading term, that is, in the Tracy-Widom distribution. We conclude with a brief discussion of the universality of these results.