  In this paper, we address a class of problems in unitary ensembles.
Specifically, we study the probability that a gap symmetric about 0, i.e.
$(-a,a)$ is found in the Gaussian unitary ensembles (GUE) and the Jacobi
unitary ensembles (JUE) (where in the JUE, we take the parameters
$\alpha=\beta$). By exploiting the even parity of the weight, a doubling of the
interval to $(a^2,\infty)$ for the GUE, and $(a^2,1)$, for the (symmetric) JUE,
shows that the gap probabilities maybe determined as the product of the
smallest eigenvalue distributions of the LUE with parameter $\alpha=-1/2,$ and
$\alpha=1/2$ and the (shifted) JUE with weights $x^{1/2}(1-x)^{\beta}$ and
$x^{-1/2}(1-x)^{\beta}$ The $\sigma$ function, namely, the derivative of the
log of the smallest eigenvalue distributions of the finite-$n$ LUE or the JUE,
satisfies the Jimbo-Miwa-Okamoto $\sigma$ form of $P_{V}$ and $P_{VI}$,
although in the shift Jacobi case, with the weight $x^{\alpha}(1-x)^{\beta},$
the $\beta$ parameter does not show up in the equation. We also obtain the
asymptotic expansions for the smallest eigenvalue distributions of the Laguerre
unitary and Jacobi unitary ensembles after appropriate double scalings, and
obtained the constants in the asymptotic expansion of the gap probablities,
expressed in term of the Barnes $G-$ function valuated at special point.
