Leveraging a general framework adapted from symbolic integration, a unified reduction-based algorithm for computing telescopers of minimal order for hypergeometric and q-hypergeometric terms has been recently developed. In this paper, we conduct a deeper exploration and put forth a new argument for the termination of the algorithm. This not only provides an independent proof of existence of telescopers, but also allows us to derive unified upper and lower bounds on the order of telescopers for hypergeometric terms and their q-analogues. Compared with known bounds in the literature, our bounds, in the hypergeometric case, are exactly the same as the tight ones obtained in 2016; while in the q-hypergeometric case, no lower bounds were known before, and our upper bound is sometimes better and never worse than the known one.

翻译：基于符号积分领域所采用的通用框架，近期已发展出一种统一化的基于归约的算法，用于计算超几何项与q-超几何项的最小阶套缩子。本文对此算法进行了更深入的探讨，并提出一种关于算法终止性的新论证。这不仅为套缩子的存在性提供了一个独立的证明，还使我们能够推导出超几何项及其q-模拟的套缩子阶数的统一上下界。与文献中已知的界相比，在超几何情形下，我们的界与2016年获得的紧界完全相同；而在q-超几何情形下，此前并无已知的下界，且我们的上界有时更优，且从不逊于已知结果。