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

TL;DR

This paper explores the complex relationships between Hankel determinants generated by discontinuous Gaussian weights and Painlevé equations, revealing new differential equations and asymptotic behaviors in orthogonal polynomials.

Contribution

It extends previous work by analyzing multiple jumps in Gaussian weights, deriving new difference and differential equations, and connecting these to Painlevé transcendents and special functions.

Findings

Derived difference and differential equations including Painlevé IV and Chazy II.

Established large n behavior of orthogonal polynomials satisfying biconfluent Heun equation.

Identified Painlevé XXXIV and a two-variable Jimbo-Miwa-Okamoto sigma form in specific scaling limits.

Abstract

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.