We consider the lim iting process that arises at the hard edge of Mutt alib-Borodin ensembles. This point process depends on $\\theta >\; 0$ and has a kernel built out o f Wright'\;s generalized Bessel functions. In a recent paper\, Claeys\, Girotti and Stivigny have established first and second order asymptotics fo r large gap probabilities in these ensembles. Thes e asymptotics take the form

\\begin{equation*}

\\mathbb{P}(\\mbox{gap on } [0\,s]) = C \\exp \\left( -a s^{2\\rho} + b s^{\\rho} + c \\ln s \\ right) (1 + o(1)) \\qquad \\mbox{as }s \\to + \\i nfty\,

\\end{equation*}

where the constant s $\\rho$\, $a$\, and $b$ have been derived explic itly via a differential identity in $s$ and the an alysis of a Riemann-Hilbert problem. Their method can be used to evaluate $c$ (with more efforts)\, but does not allow for the evaluation of $C$. In t his work\, we obtain expressions for the constants $c$ and $C$ by employing a differential identity in $\\theta$. When $\\theta$ is rational\, we find that $C$ can be expressed in terms of Barnes'\ ; $G$-function. We also show that the asymptotic f ormula can be extended to all orders in $s$. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR