Continuing previous work, this paper focuses on the summability problem of multivariate rational functions in the mixed case in which both shift and $q$-shift operators can appear. Our summability criteria rely on three ingredients including orbital decompositions, Sato's isotropy groups, and difference transformations. This work settles the rational case of the long-term project aimed at developing algorithms for symbolic summation of multivariate functions.

翻译：延续先前工作，本文聚焦于同时包含位移算子与$q$-位移算子的混合情形下多元有理函数的可和性问题。我们提出的可和性判定准则基于三个核心要素：轨道分解、Sato各向同性子群以及差分变换。本研究解决了长期项目中针对多元函数符号求和的算法开发计划的有理函数情形。