Painlevé IV, biconfluent Heun and the semi-classical Laguerre polynomials

In this paper, we investigate a certain linear statistic of the unitary ensembles with two parameters, which is equivalent to the characterization of a sequence of polynomials orthogonal with respect to a semi-classical Laguerre weight [Formula: see text] with parameters [Formula: see text], [Formula: see text], [Formula: see text]. We explore certain transformations of the recurrence coefficients and the sub-leading coefficient for the semi-classical Laguerre polynomials, which can serve as solutions of the analogs of Painlevé IV, include the Jimbo–Miwa–Okamoto [Formula: see text]-form of Painlevé IV and the discrete [Formula: see text]-form of Painlevé IV. Using Dyson’s Coulomb fluid approach, we derive the asymptotic behaviors of the recurrence coefficients and the smallest eigenvalue of large Hankel matrices generated by the semi-classical Laguerre weight. Additionally, we reduce the second-order differential equation satisfied by the orthogonal polynomial generated by the semi-classical Laguerre weight to the biconfluent Heun equation.