It is known from [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.] that the recurrence coefficients of discrete orthogonal polynomials on the nonnegative integers with hypergeometric weights satisfy a system of nonlinear difference equations. There is also a connection to the solutions of the σ-form of the sixth Painlevé equation (one of the parameters of the weights being the independent variable in the differential equation) [G. Filipuk and W. Van Assche, Discrete orthogonal polynomials with hypergeometric weights and Painlevé VI, Symmetry Integr. Geom. Methods Appl. 14 (2018), Article ID: 088, 19 pp.]. In this paper, we derive a second-order nonlinear difference equation from the system and present explicit formulas showing how this difference equation arises from the Bäcklund transformations of the sixth Painlevé equation. We also present an alternative way to derive the connection between the recurrence coefficients and the solutions of the sixth Painlevé equation.

中文翻译：

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具有超几何权重的正交多项式的递归系数的微分和差分方程以及第六个 Painlevé 方程的 Bäcklund 变换

从[G. Filipuk 和 W. Van Assche，具有超几何权重的离散正交多项式和 Painlevé VI，对称积分。几何。方法应用程序。 14(2018), 文章 ID: 088, 19 pp.] 具有超几何权重的非负整数上的离散正交多项式的递推系数满足非线性差分方程组。也与解决方案有关σ-第六Painlevé方程的形式（权重的参数之一是微分方程中的自变量）[G. Filipuk 和 W. Van Assche，具有超几何权重的离散正交多项式和 Painlevé VI，对称积分。几何。方法应用程序。 14(2018)，文章 ID：088，19 页]。在本文中，我们从系统中推导出一个二阶非线性差分方程，并给出了显示该差分方程如何从第六个 Painlevé 方程的 Bäcklund 变换产生的明确公式。我们还提出了一种替代方法来推导递归系数和第六个 Painlevé 方程的解之间的联系。