Painlev\'{e} Transcendents and the Hankel Determinants Generated by a Discontinuous Gaussian Weight

This paper studies the Hankel determinants generated by a discontinuous Gaussian weight with one and two jumps. It is an extension of Chen and Pruessner \cite{Chen2005}, in which they studied the discontinuous Gaussian weight with a single jump. By using the ladder operator approach, we obtain a series of difference and differential equations to describe the Hankel determinant for the single jump case. These equations include the Chazy II equation, continuous and discrete Painlev\'{e} IV. In addition, we consider the large $n$ behavior of the corresponding orthogonal polynomials and prove that they satisfy the biconfluent Heun equation. We also consider the jump at the edge under a double scaling, from which a Painlev\'{e} XXXIV appeared. Furthermore, we study the Gaussian weight with two jumps, and show that a quantity related to the Hankel determinant satisfies a two variables' generalization of the Jimbo-Miwa-Okamoto $\sigma$ form of the Painlev\'{e} IV.