We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at t,k=1,…,m. By employing the ladder operator approach to establish Riccati equations, we show that σ(t,…,t), the logarithmic derivative of the n-dimensional Hankel determinant, satisfies an m-variable generalization of the σ-form of a Painlevé V equation. Through investigating the Riemann–Hilbert problem for the associated orthogonal polynomials and via the Lax pair, we express σ in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and the Lax pair. In addition, when each t tends to the hard edge of the spectrum and n goes to infinity, the scaled σ is shown to satisfy a generalized σ-form of a Painlevé III equation.