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

Authors Clemens Heuberger , Daniel Krenn , Helmut Prodinger

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ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Combinatoric problems
  • Mathematics of computing → Generating functions
  • Theory of computation → Formal languages and automata theory
Keywords
  • Regular sequence
  • Mellin-Perron summation
  • summatory function
  • transducer
  • 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.

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Clemens Heuberger, Daniel Krenn, and Helmut Prodinger. Analysis of Summatory Functions of Regular Sequences: Transducer and Pascal's Rhombus. In 29th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 110, pp. 27:1-27:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.AofA.2018.27

Author Details

Clemens Heuberger
  • Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65--67, 9020 Klagenfurt am Wörthersee, Austria
Daniel Krenn
  • Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65--67, 9020 Klagenfurt am Wörthersee, Austria
Helmut Prodinger
  • Department of Mathematical Sciences, Stellenbosch University, 7602 Stellenbosch, South Africa

Funding

C. Heuberger and D. Krenn are supported by the Austrian Science Fund (FWF): P 28466-N35.

References

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