Painleve IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities

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