Painlevé VI and Hankel determinants for the generalized Jacobi Weight

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 Painlevé VI system.

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

References in corpus (3)

• Painlevé V and time dependent Jacobi polynomials
• Painleve IV and degenerate Gaussian Unitary Ensembles
• Random Matrix Theory and the Sixth Painlevé Equation

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• The lowest eigenvalue of Jacobi random matrix ensembles and Painlevé VI
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• Asymptotics of determinants of Hankel matrices via non-linear difference equations
• Painlevé V, Painlevé XXXIV and the Degenerate Laguerre Unitary Ensemble
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• Semi-classical Jacobi Polynomials, Hankel Determinants and Asymptotics
• Painlevé VI, Painlevé III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble
• Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics
• Differential and Difference Equations for Recurrence Coefficients of Orthogonal Polynomials with a Singularly Perturbed Laguerre-type Weight