Non-Holonomicity of Sequences Defined via Elementary Functions

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.

[1]  Lee A. Rubel,et al.  Some research problems about algebraic differential equations II , 1992 .

[2]  Lee A. Rubel,et al.  Some research problems about algebraic differential equations , 1983 .

[3]  Martin Klazar.  Non-holonomicity of the sequence $\log 1, \log 2, \log 3, ...$ , 2005 .

[4]  Stefan Gerhold,et al.  On Some Non-Holonomic Sequences , 2004, Electron. J. Comb..

[5]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[6]  B. F. Caviness.  On Canonical Forms and Simplification , 1970, JACM.

[7]  L. Lipshitz,et al.  D-finite power series , 1989 .

[8]  B. Ya. Levin,et al.  Entire and Meromorphic Functions , 1997 .

[9]  Henry C. Thacher,et al.  Applied and Computational Complex Analysis. , 1988 .

[10]  R. Remmert,et al.  Classical Topics in Complex Function Theory , 1997 .

[11]  Daniel Richardson,et al.  Some undecidable problems involving elementary functions of a real variable , 1969, Journal of Symbolic Logic.

[12]  Stefan Gerhold,et al.  Combinatorial Sequences: Non-Holonomicity and Inequalities , 2008 .

[13]  Marius van der Put,et al.  Galois Theory of Difference Equations , 1997 .

[14]  L. Rubel,et al.  Entire and Meromorphic Functions , 1996 .

[15]  M. K. Kerimov,et al.  Applied and computational complex analysis. Vol. 1. Power series, integration, conformal mapping, location of zeros: Henrici P. xv + 682 pp., John Wiley and Sons, Inc., New York — London, 1974☆ , 1977 .

[16]  Alfred J. van der Poorten,et al.  A Proof that Euler Missed... , 2000 .

[17]  L. Rubel,et al.  On Small Entire Functions of Exponential-Type with Given Zeros , 1961 .

[18]  G. Molteni.  Some Arithmetical Properties of the Generating Power Series for the Sequence {ζ(2k+1)}k=1∞ , 2001 .

[19]  Fritz Carlson,et al.  Über Potenzreihen mit ganzzahligen Koeffizienten , 1921 .

[20]  Y. Sibuya,et al.  The reciprocals of solutions of linear ordinary differential equations , 1985 .

[21]  Philippe Flajolet,et al.  On the Non-Holonomic Character of Logarithms, Powers, and the nth Prime Function , 2005, Electron. J. Comb..