Asymptotics of spacing distributions at the hard edge for \(\beta\)-ensembles (Q2844435)

In a previous work [J. Math. Phys. 35, No. 5, 2539--2551 (1994; Zbl 0807.60029)], the author used generalized hypergeometric functions to give a rigorous derivation of the large \(s\) asymptotic form of the general \(\beta > 0\) gap probability, under the assumption \(2/\beta \in\mathbb{Z}^+\). It this work it is shown how the details of this method can be extended to remove the requirement that \(2/\beta \in\mathbb{Z}^+\). Furthermore, a large deviation formula for the gap probability is deduced by writing it in terms of the characteristic function of a certain linear statistic. This is shown to reproduce a recent conjectured formula for a gap probability and moreover to give a prediction without the latter restriction. This extended formula, which for the constant term involves the Barnes double gamma function, is shown to satisfy an asymptotic functional equation relating the gap probability with parameters \((\beta, n, a)\) to a gap probability with parameters \((4/\beta, n', a')\), where \(n' = \beta (n + 1)/2 - 1\), \(a' = \beta (a - 2)/2 + 2\).