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Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences

Authors Clemens Heuberger , Daniel Krenn , Tobias Lechner

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Author Details

Clemens Heuberger
  • Department of Mathematics, University of Klagenfurt, Austria
Daniel Krenn
  • Fachbereich Mathematik, Paris Lodron University of Salzburg, Austria
Tobias Lechner
  • Department of Mathematics, University of Klagenfurt, Austria

Funding

  • Heuberger, Clemens: This research was funded in part by the Austrian Science Fund (FWF) [10.55776/DOC78]. For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.

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Clemens Heuberger, Daniel Krenn, and Tobias Lechner. Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.AofA.2024.24

Abstract

In the asymptotic analysis of regular sequences as defined by Allouche and Shallit, it is usually advisable to study their summatory function because the original sequence has a too fluctuating behaviour. It might be that the process of taking the summatory function has to be repeated if the sequence is fluctuating too much. In this paper we show that for all regular sequences except for some degenerate cases, repeating this process finitely many times leads to a "nice" asymptotic expansion containing periodic fluctuations whose Fourier coefficients can be computed using the results on the asymptotics of the summatory function of regular sequences by the first two authors of this paper. In a recent paper, Hwang, Janson, and Tsai perform a thorough investigation of divide-and-conquer recurrences. These can be seen as 2-regular sequences. By considering them as the summatory function of their forward difference, the results on the asymptotics of the summatory function of regular sequences become applicable. We thoroughly investigate the case of a polynomial toll function.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Regular sequence
  • Divide-and-Conquer Recurrence
  • Summatory Function
  • Asymptotic Analysis

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Supplementary Materials

References

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