Painlevé III asymptotics of Hankel determinants for a perturbed Jacobi weight

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We study the Hankel determinants associated with the weight w(x;t)=(1-x²)β(t²-x²)αh(x), x∈(-1,1), where β>-1, α+β>-1, t>1, h(x) is analytic in a domain containing [-1,1] and h(x)>0 for x∈[-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→ ∞ and t→ 1. We obtain the asymptotic approximations of the Hankel determinants, evaluated in terms of the Jimbo-Miwa-Okamoto σ-function for the Painlevé III equation. The asymptotics of the leading coefficients and the recurrence coefficients for the perturbed Jacobi polynomials are also obtained.