Hankel determinants for a perturbed Laguerre weight and Painleve V equation

TL;DR

This paper derives asymptotic formulas for Hankel determinants generated by a perturbed Laguerre weight, linking them to Painleve V equations, and analyzes related orthogonal polynomial coefficients using advanced asymptotic methods.

Contribution

It establishes a connection between Hankel determinants with perturbed Laguerre weights and Painleve V equations, providing uniform asymptotics under double scaling.

Findings

Asymptotic formulas for Hankel determinants are obtained.

Leading and recurrence coefficients of orthogonal polynomials are characterized asymptotically.

The analysis employs the Deift-Zhou nonlinear steepest descent method.

Abstract

In this paper, we study Hankel determinants generated from a perturbed Laguerre weight function, Under the double scaling scheme, we give the uniform asymptotic approximations of Hankel determinants in terms of a solution of a third-order nonlinear differential equation, which is equivalent to a particular Painleve V equation. In fact, this Painleve V equation is equivalent to the general Painleve III equation. The asymptotic approximations of the leading coefficients and the recurrence coefficients for the corresponding orthogonal polynomials also involve the painleve V equation. The asymptotic analysis is based on our earlier results by using the Deift-Zhou nonlinear steepest descent method.