Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight

Abstract

We study the Hankel determinants associated with the weight $$w(x;t)=(1-x^2{)}^{\beta }(t^2-x^2{)}^{\alpha }h(x),x\in (-1,1),$$ where $\beta >-1$, $\alpha +\beta >-1$, $t>1$, $h(x)$ is analytic in a domain containing $[-1,1]$ and $h(x)>0$ for $x\in [-1,1]$. In this paper, based on the Deift-Zhou nonlinear steepest descent analysis, we study the double scaling limit of the Hankel determinants as $n\to \infty$ and $t\to 1$. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo-Miwa-Okamoto $\sigma$-function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.

28 pages. Modifications made to the previous version arXiv:1412.8586v1