In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles w(z,t)=Aθ(z-t)e-z2+tz,here θ(x) is the Heaviside function, where A> 0 , t> 0 , and z∈ [0 , ∞). We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities R(t) and r(t) satisfy the coupled Riccati equations, from which we also prove that R(t) satisfies a particular Painlevé IV equation. Even more, σ(t) , allied to R(t) , satisfies both the discrete and continuous Jimbo–Miwa–Okamoto σ-form of the Painlevé IV equation. Finally, we consider the situation when n gets large, the second order linear differential equation satisfied by the polynomials P(x) orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation.