In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek–Jacobi type weight

By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient and the sub-leading coefficient of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of can be expressed in terms of a particular Painlevé V transcendent. The large asymptotic expansions of and are obtained by using Dyson's Coulomb fluid method together with the related difference equations. Furthermore, we study the associated Hankel determinant and show that a quantity , allied to the logarithmic derivative of , can be expressed in terms of the -function of a particular Painlevé V. The second-order differential and difference equations for are also obtained. In the end, we derive the large asymptotics of and from their relations with and .

中文翻译：

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Pollaczek-Jacobi 型正交多项式及其 Hankel 行列式的微分、差分和渐近关系

在本文中，我们研究了关于奇异扰动的 Pollaczek-Jacobi 类型权重的正交多项式

利用阶梯算子的方法，分别建立了单次正交多项式的递推系数和次导系数满足的二阶差分方程。我们表明 的对数导数可以用特定的 Painlevé V 超越数来表示。大的渐近展开并通过使用Dyson的库仑流体的方法与相关差分方程在一起而获得。此外，我们研究了相关的汉克尔行列式，并表明与 的对数导数相关的量可以表示为- 特定 Painlevé V 的函数。还获得了的二阶微分和差分方程。最后，我们得到了大量的渐近并从与他们的关系和。