Hankel Determinant for a Perturbed Laguerre Weight with Pole Singularities and Generalized Painlev\'e III' Equation

We study the Hankel determinant for the weight $x^{\alpha}{\rm exp}(-x-t_1/x-t_2/x^2), x\in[0,+\infty)$, with $\alpha>-1,~t_1\in\mathbb{R}\setminus\{0\}, ~t_2>0.$ Compared with the weight $x^{\alpha}{\rm e}^{-x-t_1/x}$ studied in prior work (where $\alpha,t_1>0$), the range of $\alpha$ in our work is extended and the parameter $t_2$ introduces a ``stronger" zero at the origin. This leads to more varied behavior of the Hankel determinant, and the interplay between $t_1$ and $t_2$ introduces uncertainty and complexity into the analysis. By using a pair of ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions, we show that the recurrence coefficients are expressed in terms of four auxiliary quantities which satisfy a system of difference equations that can be iterated. We also establish two coupled second order partial differential equations (PDEs) satisfied by two of the auxiliary quantities, which are reduced to a Painlev\'{e} III$^\prime$ equation when $t_2\rightarrow0^+$. Moreover, the logarithmic derivative of the Hankel determinant is shown to satisfy a second order six degree PDE which is reduced to the $\sigma$-form of the Painlev\'{e} III$^{\prime}$ equation when $t_2\rightarrow0^+$. Under suitable double scaling, we obtain the limiting forms of the above PDEs and deduce the equilibrium density of the eigenvalues for the unitary ensemble. We extend our analysis to the Hankel determinant for $x^{\alpha}\exp(-x-\sum_{k=1}^m t_k/x^k)$ with $m=3$. For general $m$, we outline a derivation that leads, at least in principle, to the PDE satisfied by the logarithmic derivative of the Hankel determinant.