Laguerre unitary ensembles with jump discontinuities, PDEs and the coupled Painlevé V system

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