Summary: C. A. Tracy and H. Widom [Commun. Math. Phys. 159, 151–174 (1994; Zbl 0789.35152))] have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PTV and PII transcendent respectively. We generalise these results to the evaluation of
where for and otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of
Of particular interest are and , and their scaled limits, which gives the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto -function theory of PIV and PII [cf, K. Okamoto, [Studies on the Painlevé Equations. III. Second and Fourth Painlevé Equations, and . Math. Ann. 275, 221–255 (1986; Zbl 0589.58008)], for which we give a self contained presentation based on the recent work of M. Noumi and Y. Yamada [Nagoya Math. J. 153, 53–86 (1999; Zbl 0932.34088)]. We point out that the same approach can be used to study the quantities and for the other classical matrix ensembles.