The recurrence coefficients of a semi-classical Laguerre polynomials and the large n asymptotics of the associated Hankel determinant

Abstract

In this paper, we study the recurrence coefficients of a deformed or semi-classical Laguerre polynomials orthogonal with respect to the weight [Formula: see text] Here [Formula: see text], [Formula: see text] and [Formula: see text]. We will describe this problem in terms of the ratio [Formula: see text] where ultimately [Formula: see text] is bounded away from 0, and close to 1. From the ladder operator approach, and the associated compatibility conditions, the recurrence coefficients satisfy a second order ordinary differential equation (ODE) when viewed as functions of the parameter [Formula: see text] in the weight. Then we work out the large degree asymptotics of their recurrence coefficients. We also discuss the associated Hankel determinant. We show that the logarithmic derivative of [Formula: see text] can be expressed in terms of the recurrence coefficients, and obtained the large degree asymptotics of [Formula: see text]. Based on this result, we compute the probability that an [Formula: see text] (or [Formula: see text]) random matrix from a generalized Gaussian Unitary Ensemble (gGUE) is positive definite.