Perturbed Hankel determinant, correlation functions and Painlevé equations

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We continue with the study of the Hankel determinant, Dₙ(t,α,β):=(∫₀¹xʲ⁺ᵏw(x;t,α,β)dx)ⱼ,ₖ₌₀ⁿ⁻¹, generated by a Pollaczek-Jacobi type weight, w(x;t,α,β):=xα(1-x)β e⁻ᵗ/ˣ, x∈ [0,1], α>0, β>0, t≥ 0. 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 5 (P5). In fact the logarithmic of the Hankel determinant has an integral representation in terms of a particular P5. 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:=2n²t is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular P3'. 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 P5, %which can be degenerate to a particular P3 and its small and large s asymptotic expansions are also found.