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 w(x,t)=xαe−x−tx2,x∈(0,∞),α>0,t≥0. 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 Rn(t) and rn(t) by using the Ladder operator approach. We show that the Hankel determinant has an integral representation in terms of Rn(t), where the quantity Rn(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 σn(t)=−∑j=0n−1Rj(t), 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.