  In this paper, we study the Hankel determinant associated with the degenerate
Laguerre unitary ensemble. This problem originates from the largest or smallest
eigenvalue distribution of the degenerate Laguerre unitary ensemble. We derive
the ladder operators and its compatibility condition with respect to a general
perturbed weight. By applying the ladder operators to our problem, we obtain
two auxiliary quantities $R_n(t)$ and $r_n(t)$ and show that they satisfy the
coupled Riccati equations, from which we find that $R_n(t)$ satisfies the
Painlev\'{e} V equation. Furthermore, we prove that $\sigma_{n}(t)$, a quantity
related to the logarithmic derivative of the Hankel determinant, satisfies both
the continuous and discrete Jimbo-Miwa-Okamoto $\sigma$-form of the
Painlev\'{e} V. In the end, by using Dyson's Coulomb fluid approach, we
consider the large $n$ asymptotic behavior of our problem at the soft edge,
which gives rise to the Painlev\'{e} XXXIV equation.
