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In this paper, we consider the Hankel determinants associated with thesingularly perturbed Laguerre weight $w(x)=x^\alpha e^{-x-t/x}$, $x\in (0,\infty)$, $t>0$ and $\alpha>0$. When the matrix size $n\to\infty$, we obtain anasymptotic formula for the Hankel determinants, valid uniformly for $t\in (0,d]$, $d>0$ fixed. A particular Painlev\'{e} III transcendent is involved in theapproximation, as well as in the large-$n$ asymptotics of the leadingcoefficients and recurrence coefficients for the corresponding perturbedLaguerre polynomials. The derivation is based on the asymptotic results in anearlier paper of the authors, obtained by using the Deift-Zhou nonlinearsteepest descent method.

Painlev\'e III asymptotics of Hankel determinants for a singularly perturbed Laguerre weight