  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.
