Perturbed Hankel determinant, correlation functions and Painlev\'e equations

We continue with the study of the Hankel determinant, $$D_{n}(t,\alpha,\beta):=\det\left(\int_{0}^{1}x^{j+k}w(x;t,\alpha,\beta)dx\right)_{j,k=0}^{n-1},$$ generated by a Pollaczek-Jacobi type weight, $$w(x;t,\alpha,\beta):=x^{\alpha}(1-x)^{\beta}{\rm e}^{-t/x}, \quad x\in [0,1], \quad \alpha>0, \quad \beta>0, \quad t\geq 0.$$ This reduces to the "pure" Jacobi weight at $t=0.$ We may take $\alpha\in \mathbb{R}$, in the situation while $t$ is strictly greater than $0.$ It was shown in Chen and Dai (2010), that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamoto $\sigma$-form of Painlev\'e \uppercase\expandafter{\romannumeral5} ({\rm P_{\uppercase\expandafter{\romannumeral5}}}). In fact the logarithmic of the Hankel determinant has an integral representation in terms of a particular {\rm P_{\uppercase\expandafter{\romannumeral5}}}. \\ In this paper, we show that, under a double scaling, where $n$ the dimension of the Hankel matrix tends to $\infty$, and $t$ tends to $0^{+},$ such that $s:=2n^2t$ is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular {\rm P_{\uppercase\expandafter{\romannumeral3}'}}. Expansions of the scaled Hankel determinant for small and large $s$ are found. A further double scaling with $\alpha=-2n+\lambda,$ where $n\rightarrow \infty$ and $t,$ tends to $0^{+},$ such that $s:=nt$ is finite. In this situation the scaled Hankel determinant has an integral representation in terms of a particular {\rm P_{\uppercase\expandafter{\romannumeral5}}}, %which can be degenerate to a particular {\rm P_{\uppercase\expandafter{\romannumeral3}}} and its small and large $s$ asymptotic expansions are also found.