A singular linear statistic for a perturbed LUE and the Hankel matrices

In this paper, we investigate the Hankel determinant generated by a singular Laguerre weight with two parameters. Using ladder operators adapted to monic orthogonal polynomials associated with the weight, we show that one of the auxiliary quantities is a solution to the Painlevé III′ equation and derive the discrete σ-forms of two logarithmic partial derivatives of the Hankel determinant. We approximate the second-order differential equation satisfied by the monic orthogonal polynomials with respect to the singular Laguerre weight with two parameters to the double confluent Heun equation, leveraging the scaling limit for two parameters and the dimension of the Hankel determinant. In addition, we establish the asymptotic behavior of the smallest eigenvalue of large Hankel matrices associated with the weight with two parameters, using the Coulomb fluid method and the Rayleigh quotient.