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 w(z; s, t) = z(alpha)e(-sz2 + tz), z is an element of [0, infinity), with parameters alpha > 0, s > 0, t is an element of & Ropf;. 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 Painlev & eacute; IV, include the Jimbo-Miwa-Okamoto sigma-form of Painlev & eacute; IV and the discrete sigma-form of Painlev & eacute; 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.