  We study the Hankel determinant generated by the Laguerre weight with jump
discontinuities at $t_k, k=1,\cdots,m$. By employing the ladder operator
approach to establish Riccati equations, we show that
$\sigma_n(t_1,\cdots,t_m)$, the logarithmic derivative of the $n$-dimensional
Hankel determinant, satisfies a generalization of the $\sigma$-from of
Painlev\'{e} V equation. Through investigating the Riemann-Hilbert problem for
the associated orthogonal polynomials and via Lax pair, we express $\sigma_n$
in terms of solutions of a coupled Painlev\'{e} V system. We also build
relations between the auxiliary quantities introduced in the above two methods,
which provides connections between the Riccati equations and Lax pair. In
addition, when each $t_k$ tends to the hard edge of the spectrum and $n$ goes
to $\infty$, the scaled $\sigma_n$ is shown to satisfy a generalized
Painlev\'{e} III equation.
