  This paper studies the Hankel determinant generated by a perturbed Jacobi
weight, which is closely related to the largest and smallest eigenvalue
distribution of the degenerate Jacobi unitary ensemble. By using the ladder
operator approach for the orthogonal polynomials, we find that the logarithmic
derivative of the Hankel determinant satisfies a nonlinear second-order
differential equation, which turns out to be the Jimbo-Miwa-Okamoto
$\sigma$-form of the Painlev\'{e} VI equation by a translation transformation.
We also show that, after a suitable double scaling, the differential equation
is reduced to the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlev\'{e} III. In
the end, we obtain the asymptotic behavior of the Hankel determinant as
$t\rightarrow1^{-}$ and $t\rightarrow0^{+}$ in two important cases,
respectively.
