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Non-Holonomicity of Sequences Defined via Elementary Functions
Jason P. Bell1, Stefan Gerhold2, Martin Klazar3, and Florian Luca4
1Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, B.C. V5A 1S6, Canada
jpb@sfu.ca
2Christian Doppler Laboratory for Portfolio Risk Management, Vienna University of Technology, Wiedner Hauptstraße 8/105-1 A-1040 Vienna, Austria
sgerhold@fam.tuwien.ac.at
3Department 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
klazar@kam.mff.cuni.cz
4Instituto de Matem´aticas UNAM, C.P. 58 089 Campus Morelia, Michoacán, Mexico
fluca@matmor.unam.mx
Annals of Combinatorics 12 (1) p.1-16 March, 2008
AMS Subject Classification:11B37, 11B83
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 logn. 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.
Keywords: holonomic sequences, fewnomials, meromorphic functions

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