Fuchsian holonomic sequences

Abstract

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.