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

We continue with the study of the Hankel determinant, $$D_n(t,\alpha ,\beta ):=\det {({\int }_0^1{x}^{j+k}w(x;t,\alpha ,\beta )dx)}_{j,k=0}^{n-1},$$ generated by a Pollaczek-Jacobi type weight, $$w(x;t,\alpha ,\beta ):=x^{\alpha }(1-x{)}^{\beta }{\mathrm{e}}^{-t\mathrm{/}x},x\in [0,1],\alpha >0,\beta >0,t\ge 0.$$ 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 $\mathrm{\infty }$, and $t$ tends to $0^{+},$ such that $s:=2n^{2}t$ 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 $n\to \mathrm{\infty }$ 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.