{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/asymptotic-analysis-of-regular-sequences","title":"Asymptotic Analysis of Regular Sequences","arxiv_id":"1810.13178","date":"2018-10-31","proceeding":null,"authors":["Clemens Heuberger","Daniel Krenn"],"abstract":"In this article, $q$-regular sequences in the sense of Allouche and Shallit are analysed asymptotically. It is shown that the summatory function of a regular sequence can asymptotically be decomposed as a finite sum of periodic fluctuations multiplied by a scaling factor. Each of these terms corresponds to an eigenvalue of the sum of matrices of a linear representation of the sequence; only the eigenvalues of absolute value larger than the joint spectral radius of the matrices contribute terms which grow faster than the error term. The paper has a particular focus on the Fourier coefficients of the periodic fluctuations: They are expressed as residues of the corresponding Dirichlet generating function. This makes it possible to compute them in an efficient way. The asymptotic analysis deals with Mellin--Perron summations and uses two arguments to overcome convergence issues, namely H\\\"older regularity of the fluctuations together with a pseudo-Tauberian argument. Apart from the very general result, three examples are discussed in more detail: sequences defined as the sum of outputs written by a transducer when reading a $q$-ary expansion of the input; the amount of esthetic numbers in the first~$N$ natural numbers; and the number of odd entries in the rows of Pascal's rhombus. For these examples, very precise asymptotic formul\\ae{} are presented. In the latter two examples, prior to this analysis only rough estimates were known.","url_abs":"http://arxiv.org/abs/1810.13178v4","url_pdf":"http://arxiv.org/pdf/1810.13178v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"asymptotic-analysis-of-regular-sequences","repo_url":"https://gitlab.com/dakrenn/regular-sequence-fluctuations","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}