This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second‐order differential equation, which turns out to be the Jimbo–Miwa–Okamoto σ‐form of the Painlevé VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced to the Jimbo–Miwa–Okamoto σ‐form of the Painlevé III. In the end, we obtain the asymptotic behavior of the Hankel determinant as t→1⁻ and t→0⁺ in two important cases, respectively.

中文翻译：

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PainlevéVI，PainlevéIII和汉高行列式与退化的Jacobi ary合奏相关

本文研究了由扰动的Jacobi权重生成的Hankel行列式，它与退化的Jacobi ary系综的最大和最小特征值分布密切相关。通过使用用于正交多项式梯操作的方法，我们发现，对汉克尔行列式满足的对数导数一二阶非线性微分方程，这原来是神保三轮-冈本σ -形式的PAINLEVE VI方程由翻译转换。我们还表明，中，合适的双缩放之后，所述微分方程降低到神保三轮-冈本σ -形式的PAINLEVE III的。最后，我们得到汉克尔行列式作为渐近行为吨→1 ^-和在两种重要情况下，t →0 ⁺。