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

We study the Hankel determinant generated by a deformed Hermite weight with one jump w(z,t,gamma)=e(-z2+tz)|z-t|(gamma)(A+B theta(z-t)), where A >= 0, A + B >= 0, t is an element of R, gamma > -1, and z is an element of 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 alpha(n), beta(n), R-n(t), and r(n)(t). In particular, we find that the auxiliary quantities R-n(t) and r(n)(t) satisfy the coupled Riccati equations, and R-n(t) satisfies a particular Painleve IV equation. Based on the above results, we show that sigma(n)(t) and (sigma) double under bar (n)(t), two quantities related to the Hankel determinant and R-n(t), satisfy the continuous and discrete sigma-form equations, respectively. In the end, we also discuss the large n asymptotic behavior of R-n(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.