We continue with the study of the Hankel determinant, D n (t, α, β) := det , generated by a Pollaczek-Jacobi type weight,This reduces to the "pure" Jacobi weight at t = 0. We may take α ∈ 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 σ -form of Painlevé V ( P V ). In fact the logarithmic of the Hankel determinant has an integral representation in terms of a particular P V . In this paper, we show that, under a double scaling, where n the dimension of the Hankel matrix tends to ∞ , 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 P III ′ . Expansions of the scaled Hankel determinant for small and large s are found. A further double scaling with α = −2n + λ, where n → ∞ 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 P V , and its small and large s asymptotic expansions are also found. The reproducing kernel in terms of monic polynomials orthogonal with respect to the Pollaczek-Jacobi type weight, under the origin (or hard edge) scaling may be expressed in terms of the solutions of a second order linear ordinary differential equation (ODE). * chenminfst@gmail.com † Corresponding author(Yang Chen), yayangchen@umac.mo and yangbrookchen@yahoo.co.uk ‡ faneg@fudan.edu.cn 1 With special choices of the parameters, the limiting (double scaled) kernel and the second order ODE degenerate to Bessel kernel and the Bessel differential equation, respectively. We also applied this method to polynomials orthogonal with respect to the perturbed Laguerre weight; w(x; t, α) := x α e −x e −t/x , 0 ≤ x < ∞, α > 0, t > 0. The scaled kernel at origin of this perturbed Laguerre ensemble has the same behavior with the above limiting kernel, although difference scaled schemes are adopted on these two kernels.2