Hankel determinants for a singular complex weight and the first and third Painlevé transcendents

Shuai-Xia Xu, Dan Dai, Yu-Qiu Zhao

In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight $e^{-n(z-\log z+t/z)}$ for $t<0$ and $z$ on certain contours in the complex pl…