Painlevé VI and Hankel determinants for the generalized Jacobi Weight

Details Painlevé VI and Hankel determinants for the generalized Jacobi Weight Painlevé VI and Hankel determinants for the generalized Jacobi Weight

2009-08-04

arxiv.org/abs/0908.0558v2

Mathematics

Classical Analysis and ODEs

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

Scientific paper

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.

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