FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 5, PAGES 31-39

Remarks on linear independence of $q$-harmonic series

P. Bundschuh

Abstract

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For any rational integer $q$, $|q|\; >\; 1$, the linear independence over $$Q of the numbers $1$, $$z_q(1), and $$z_-q(1) is proved; here $$z_q(1) = å_n=1^¥1/(qⁿ - 1) is so-called $q$-harmonic series or $q$-zeta-value at the point $1$. Besides this, a measure of linear independence of these numbers is established.

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