  We study the Hankel determinant generated by a deformed Hermite weight with
one jump $w(z,t,\gamma)=e^{-z^2+tz}|z-t|^{\gamma}(A+B\theta(z-t))$, where
$A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ and $z\in\textbf{R}$. 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 $r_n(t)$. Especially, 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 Painlev\'{e} IV equation. Based on above
results, we show that $\sigma_n(t)$ and $\hat{\sigma}_n(t)$, 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.
