Many sequences that arise in combinatorics and the analysis of algorithms turn out to be holonomic (note that some authors prefer the terminology D-finite). In this paper, we study various basic algorithmic problems for such sequences \((f_n)_{n \in {\mathbb {N}}}\): how to compute their asymptotics for large n? How to evaluate \(f_n\) efficiently for large n and/or large precisions p? How to decide whether \(f_n > 0\) for all n? We restrict our study to the case when the generating function \(f = \sum _{n \in {\mathbb {N}}} f_n z^n\) satisfies a Fuchsian differential equation (often it suffices that the dominant singularities of f be Fuchsian). Even in this special case, some of the above questions are related to long-standing problems in number theory. We will present algorithms that work in many cases and we carefully analyze what kind of oracles or conjectures are needed to tackle the more difficult cases.

组合学和算法分析中出现的许多序列被证明是完整的（请注意，一些作者更喜欢术语 D-有限）。在本文中，我们研究此类序列\((f_n)_{n \in {\mathbb {N}}}\)的各种基本算法问题：如何计算大n的渐近线？如何针对大n和/或大精度p有效评估\(f_n\)？如何判断所有n是否满足\(f_n > 0\)？我们将我们的研究限制在生成函数\(f = \sum _{n \in {\mathbb {N}}} f_n z^n\)满足 Fuchsian 微分方程的情况（通常只要f为 Fuchsian）。即使在这种特殊情况下，上述一些问题也与数论中长期存在的问题有关。我们将提出适用于多种情况的算法，并仔细分析需要什么样的预言或猜想来解决更困难的情况。