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Remarks on linear independence of q-harmonic series

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Abstract

For any rational integer q, |q| > 1, the linear independence over \( \mathbb{Q} \) of the numbers 1, ζ q (1), and ζ −q (1) is proved; here \( {\zeta_q}(1) = \sum\limits_{n = 1}^\infty {\frac{1}{{{q^n} - 1}}} \) is the so-called q-harmonic series or the q-zeta-value at the point 1. Besides this, a measure of linear independence of these numbers is established.

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Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Weyertal 86-90, D-50931, Köln, Germany

    P. Bundschuh

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  1. P. Bundschuh
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Correspondence to P. Bundschuh.

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Dedicated to the memory of Alexander O. Gelfond

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 16, No. 5, pp. 31–39, 2010.

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Bundschuh, P. Remarks on linear independence of q-harmonic series. J Math Sci 180, 550–555 (2012). https://doi.org/10.1007/s10958-012-0653-2

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