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Non-Holonomicity of Sequences Defined via Elementary Functions

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Abstract.

We present a new method for proving non-holonomicity of sequences, which is based on results about the number of zeros of elementary and of analytic functions. Our approach is applicable to sequences that are defined as the values of an elementary function at positive integral arguments. We generalize several recent results, e.g., non-holonomicity of the logarithmic sequence is extended to rational functions involving log n. Moreover, we show that the sequence that arises from evaluating the Riemann zeta function at an increasing integer sequence with bounded gap lengths is not holonomic.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, B.C., V5A 1S6, Canada

    Jason P. Bell

  2. Christian Doppler Laboratory for Portfolio Risk Management, Vienna University of Technology, Wiedner Hauptstraße 8/105-1, A-1040, Vienna, Austria

    Stefan Gerhold

  3. Department of Applied Mathematics (KAM) and Institute for Theoretic Computer Science (ITI), Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00, Praha, Czech Republic

    Martin Klazar

  4. Instituto de Matemáticas UNAM, C.P. 58 089 Campus Morelia, Michoaćan, Mexico

    Florian Luca

Authors
  1. Jason P. Bell
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  2. Stefan Gerhold
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  3. Martin Klazar
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  4. Florian Luca
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Corresponding author

Correspondence to Jason P. Bell.

Additional information

Received May 5, 2006

Stefan Gerhold: Supported in part by the Christian Doppler Research Association (CDG). S. Gerhold gratefully acknowledges a fruitful collaboration and continued support by Bank Austria and ÖBFA through CDG. Supported in part by the FWF grant F1305.

Martin Klazar: ITI is supported as project 1M0021620808 by Ministry of Education of the Czech Republic.

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Bell, J.P., Gerhold, S., Klazar, M. et al. Non-Holonomicity of Sequences Defined via Elementary Functions. Ann. Comb. 12, 1–16 (2008). https://doi.org/10.1007/s00026-008-0333-6

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  • DOI: https://doi.org/10.1007/s00026-008-0333-6

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