Painlevé IV, σ-form, and the deformed Hermite unitary ensembles

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We study the Hankel determinant generated by a deformed Hermite weight with one jump w(z,t,γ)=e−z2+tz|z−t|γ(A+Bθ(z−t)), where A ≥ 0, A + B ≥ 0, t ∈ R, γ > −1, and z ∈ R. By using the ladder operators for the corresponding monic orthogonal polynomials and their relative compatibility conditions, we obtain a series of difference and differential equations to describe the relations among αn, βn, Rn(t), and rn(t). In particular, we find that the auxiliary quantities Rn(t) and rn(t) satisfy the coupled Riccati equations, and Rn(t) satisfies a particular Painlevé IV equation. Based on the above results, we show that σn(t) and σ^n(t), two quantities related to the Hankel determinant and Rn(t), satisfy the continuous and discrete σ-form equations, respectively. In the end, we also discuss the large n asymptotic behavior of Rn(t), which produce the expansion of the logarithmic of the Hankel determinant and the asymptotic of the second order differential equation of the monic orthogonal polynomials.