Asymptotics of spacing distributions at the hard edge for $\beta$-ensembles

In a previous work [J. Math. Phys. {\bf 35} (1994), 2539--2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large $s$ asymptotic form of the general $\beta > 0$ gap probability $E_\beta^{\rm hard}(0;(0,s);\beta a/2)$, provided both $\beta a /2 \in \mathbb Z_\ge 0$ and $2/\beta \in \mathbb Z^+$. It 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 $E_\beta(n;(0,x);{\rm ME}_{\beta,N}(\lambda^{a \beta /2} e^{\beta N \lambda/2}))$ is deduced by writing it in terms of the charateristic function of a certain linear statistic. By scaling $x = s/(4N)^2$ and taking $N \to \infty$, this is shown to reproduce a recent conjectured formula for $E_\beta^{\rm hard}(n;(0,s);\beta a/2)$, $\beta a /2 \in \mathbb Z_{\ge 0}$, 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 $n'=\beta(n+1)/2-1$, $a'=\beta(a-2)/2+2$.