License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2018.27
URN: urn:nbn:de:0030-drops-89204
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8920/
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Heuberger, Clemens ; Krenn, Daniel ; Prodinger, Helmut

Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus

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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 multiplied by a scaling factor. Each summand corresponds to an 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.

BibTeX - Entry

@InProceedings{heuberger_et_al:LIPIcs:2018:8920,
  author =	{Clemens Heuberger and Daniel Krenn and Helmut Prodinger},
  title =	{{Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus}},
  booktitle =	{29th International Conference on Probabilistic,  Combinatorial and Asymptotic Methods for the Analysis of Algorithms  (AofA 2018)},
  pages =	{27:1--27:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-078-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{110},
  editor =	{James Allen Fill and Mark Daniel Ward},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8920},
  URN =		{urn:nbn:de:0030-drops-89204},
  doi =		{10.4230/LIPIcs.AofA.2018.27},
  annote =	{Keywords: Regular sequence, Mellin-Perron summation, summatory function, transducer, Pascal's rhombus}
}

Keywords: Regular sequence, Mellin-Perron summation, summatory function, transducer, Pascal's rhombus
Collection: 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018)
Issue Date: 2018
Date of publication: 18.06.2018

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