Geodesic
Remarks on linear independence of $q$-harmonic series
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 5, pp. 31-39

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For any rational integer $q$, $|q|>1$, the linear independence over $\mathbb Q$ of the numbers $1$, $\zeta_q(1)$, and $\zeta_{-q}(1)$ is proved; here $\zeta_q(1)=\sum_{n=1}^\infty\frac1{q^n-1}$ 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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     author = {P. Bundschuh},
     title = {Remarks on linear independence of $q$-harmonic series},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {31--39},
     publisher = {mathdoc},
     volume = {16},
     number = {5},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2010_16_5_a2/}
}
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P. Bundschuh. Remarks on linear independence of $q$-harmonic series. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 5, pp. 31-39. http://geodesic.mathdoc.fr/item/FPM_2010_16_5_a2/