The Hankel determinant for a semiclassical Laguerre unitary ensemble, Painlevé IV and Heun equations

We analyze the asymptotic behavior of the Hankel determinant generated by a semiclassical Laguerre weight. For this, we use ladder operators and track the evolution of parameters to establish that an auxiliary quantity associated with the semiclassical Laguerre weight satisfies the Painlevé IV equation, subject to suitable transformations of variables. Using the Coulomb fluid method, we derive the large-$n$ expansion of the logarithmic form of the Hankel determinant. This allows us to gain insights into the scaling and fluctuations of the determinant, providing a deeper understanding of its behavior in the semiclassical Laguerre ensemble. Moreover, we delve into the asymptotic evaluation of monic orthogonal polynomials with respect to the semiclassical Laguerre weight, focusing on a special case. In doing so, we shed light on the properties and characteristics of these polynomials in the context of the ensemble. Furthermore, we explore the relation between the second-order differential equations satisfied by the monic orthogonal polynomials with respect to the semiclassical Laguerre weight and the tri-confluent Heun equations or the bi-confluent Heun equations.