{"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/analysis-of-summatory-functions-of-regular","title":"Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus","arxiv_id":"1802.03266","date":"2018-02-09","proceeding":null,"authors":["Clemens Heuberger","Daniel Krenn","Helmut Prodinger"],"abstract":"The summatory function of a $q$-regular sequence in the sense of Allouche and Shallit is analysed asymptotically. The result is a sum of periodic fluctuations for eigenvalues of absolute value larger than the joint spectral radius of the matrices of a linear representation of the sequence. The Fourier coefficients of the fluctuations are expressed in terms of residues of the corresponding Dirichlet generating function. A known pseudo Tauberian argument is extended in order to overcome convergence problems in Mellin--Perron summation. Two examples are discussed in more detail: The case of sequences defined as the sum of outputs written by a transducer when reading a $q$ary expansion of the input and the number of odd entries in the rows of Pascal's rhombus.","url_abs":"https://arxiv.org/abs/1802.03266v1","url_pdf":"https://arxiv.org/pdf/1802.03266v1.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":"analysis-of-summatory-functions-of-regular","repo_url":"https://gitlab.com/dakrenn/regular-sequence-fluctuations","is_official":0,"mentioned_in_paper":0,"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}