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arXiv · 0908.0558

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

Dan Dai·Lun Zhang
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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.

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Dan Dai, Lun Zhang. 2009-08-04. Painlev\'e VI and Hankel determinants for the generalized Jacobi Weight. https://doi.org/10.1088/1751-8113/43/5/055207

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