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Linear Independence of q-Analogues of Certain Classical Constants

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Abstract

Let K be ℚ or an imaginary quadratic number field, and q ∈ K an integer with ¦q¦ > 1. We give a quantitative version of Σn≥1 an/(qn − 1) ∉ K for non-zero periodic sequences (an) in K of period length ≤ 2. As a corollary, we get a quantitative version of the linear independence over K of 1, the q-harmonic series, and a q-analogue of log 2. A similar result on 1, the q-harmonic series, and a q-analogue of ζ(2) is also proved. Mathematics Subject Classification (2000): 11J72, 11J82

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Authors and Affiliations

  1. Department of Mathematics, University of Cologne, Weyertal 86-90, 50931, Köln, Germany

    Peter Bundschuh

  2. Department of Mathematical Sciences, University of Oulu, P.O.Box 3000, 90114, Oulu, Finland

    Keijo Väänänen

Authors
  1. Peter Bundschuh
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  2. Keijo Väänänen
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Bundschuh, P., Väänänen, K. Linear Independence of q-Analogues of Certain Classical Constants. Results. Math. 47, 33–44 (2005). https://doi.org/10.1007/BF03323010

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