In this paper, we construct a new hierarchy based on the third q-discrete Painlevé equation (qPIII) and also study the hierarchy of the second q-discrete Painlevé equation (qPII). Both hierarchies are derived by using reductions of the general lattice modified Korteweg-de Vries equation. Our results include Lax pairs for both hierarchies and these turn out to be higher degree expansions of the non-resonant ones found by Joshi and Nakazono [29] for the second-order cases. We also obtain Bäcklund transformations for these hierarchies. Special q-rational solutions of the hierarchies are constructed and corresponding q-gamma functions that solve the associated linear problems are derived.

中文翻译：

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q-离散Painlevé方程的层次

在本文中，我们基于第三个 q-离散 Painlevé 方程 (qPIII) 构建了一个新的层次结构，并研究了第二个 q-离散 Painlevé 方程 (qPII) 的层次结构。两个层次都是通过使用一般晶格修正 Korteweg-de Vries 方程的约简推导出来的。我们的结果包括两个层次结构的 Lax 对，这些结果是 Joshi 和 Nakazono [29] 在二阶情况下发现的非共振对的更高程度的扩展。我们还获得了这些层次结构的 Bäcklund 变换。构建了层次结构的特殊 q-有理解，并导出了解决相关线性问题的相应 q-gamma 函数。