Painlevé VI and Hankel determinants for the generalized Jacobi Weight

Abstract

We study the Hankel determinant of the generalized Jacobi weight $(x-t)^γx^α(1-x)^β$ for $x\in[0,1]$ with $α, β>0$, $t < 0 $ and $γ\in\mathbb{R}$. 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 $τ$-function for the Painlevé VI system.

20 pages. Revised version with some modifications. Typos corrected, reference updated