Painlev\'e VI and Hankel determinants for the generalized Jacobi Weight

TL;DR

This paper investigates the Hankel determinant associated with a generalized Jacobi weight, revealing its connection to the Painlevé VI system through ladder operators and $ au$-functions.

Contribution

It establishes a novel link between Hankel determinants for generalized Jacobi weights and Painlevé VI equations using ladder operator techniques.

Findings

Logarithmic derivative of Hankel determinant characterized by Painlevé VI $ au$-function.

Derived explicit relations between orthogonal polynomials and Painlevé equations.

Enhanced understanding of special function connections in orthogonal polynomial theory.

Abstract

We study the Hankel determinant of the generalized Jacobi weight $(x-t)^{\gamma}x^\alpha(1-x)^\beta$ for $x\in[0,1]$ with $\alpha, \beta>0$, $t < 0 $ and $\gamma\in\mathbb{R}$. Based on the ladder operators for the corresponding monic orthogonal polynomials $P_n(x)$, it is shown that the logarithmic derivative of Hankel determinant is characterized by a $\tau$-function for the Painlev\'e VI system.