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DOI: 10.4230/LIPIcs.AofA.2024.24
Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences

Authors: Clemens Heuberger, Daniel Krenn, and Tobias Lechner

Published in: LIPIcs, Volume 302, 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)

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.
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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)

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@InProceedings{heuberger_et_al:LIPIcs.AofA.2024.24,
  author =	{Heuberger, Clemens and Krenn, Daniel and Lechner, Tobias},
  title =	{{Analysis of Regular Sequences: Summatory Functions and Divide-And-Conquer Recurrences}},
  booktitle =	{35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024)},
  pages =	{24:1--24:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-329-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{302},
  editor =	{Mailler, C\'{e}cile and Wild, Sebastian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.AofA.2024.24},
  URN =		{urn:nbn:de:0030-drops-204597},
  doi =		{10.4230/LIPIcs.AofA.2024.24},
  annote =	{Keywords: Regular sequence, Divide-and-Conquer Recurrence, Summatory Function, Asymptotic Analysis}
}
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