Asymptotic relations for semi-classical Laguerre orthogonal polynomials and the associated Hankel determinants

Abstract

We study recurrence coefficients of semi-classical Laguerre orthogonal polynomials and the associated Hankel determinant generated by a semi-classical Laguerre weight [Formula: see text]. If t = 0, it is reduced to the classical Laguerre weight. For t > 0, this weight tends to zero faster than the classical Laguerre weight as x → ∞. In the finite n-dimensional case, we obtain two auxiliary quantities R n( t) and r n( t) by using the Ladder operator approach. We show that the Hankel determinant has an integral representation in terms of R n( t), where the quantity R n( t) is closely related to a second-order nonlinear differential equation. Furthermore, we derive a second-order nonlinear differential equation and also a second-order differential equation for the auxiliary quantity [Formula: see text], which is also related to the logarithmic derivative of the Hankel determinant. In the infinite n-dimensional case, we consider the asymptotic behaviors of the recurrence coefficients and the scaled Laguerre orthogonal polynomials by using the Coulomb fluid method.