In the context of multivariate statistical theory the paper offers
a formula (or rather two formulae) for the distribution of the
maximal or minimal eigenvalue of certain beta-dependent random
matrices at any positive parameter beta. For beta equal to 1 the
well known real-matrix distribution is reproduced, for beta equal
to 2 and 4 its extension to complex and quaternionic matrix
ensambles is obtained.

The method is found in the Selberg-type integration of the
eigenvalue densities leading to the hypergeometric functions of
matrix argument. Remarkably related also to  symmetric functions,
Calogero-Sutherland models and Jack polynomials. Allowing, in
addition, for an efficient numerical evaluation (cf. refs. [12.
13} by Koev et al) sampled by two Figures which demonstrate
agreement with empirical data generated from 10000 samples at beta
= 2 (complex matrices) and, in a terminating case, at a growing
beta, respectively.


MR2399565 Dumitriu, Ioana; Koev, Plamen Distributions of the
extreme eigenvalues of beta-Jacobi random matrices. SIAM J. Matrix
Anal. Appl. 30 (2008), no. 1, 1--6. 65F15 (15A52 33C70 60F05
62H10)