  We study the distribution of the largest eigenvalue in the "Pfaffian"
classical ensembles of random matrix theory, namely in the Gaussian orthogonal
(GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical
skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar
(NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of
Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian"
identities due to Knuth, we explicitly construct these semi-classical
skew-orthogonal polynomials in terms of the semi-classical orthogonal
polynomials studied by NM in the case of the GUE. With these polynomials we
obtain expressions for the cumulative distribution functions of the largest
eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis
of these skew-orthogonal polynomials in the limit of large matrix size, we
obtain an alternative derivation of the Tracy-Widom distributions for GOE and
GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof
of which employs the characterization of Pfaffians in terms of perfect
matchings and link diagrams.
