Geodesic
On the irrationality measure for a~$q$-analogue of $\zeta(2)$
Sbornik. Mathematics, Tome 193 (2002) no. 8, pp. 1151-1172

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A Liouville-type estimate is proved for the irrationality measure of the quantities $$ \zeta_q(2) =\sum_{n=1}^\infty\frac{q^n}{(1-q^n)^2} $$ with $q^{-1}\in\mathbb Z\setminus\{0,\pm1\}$. The proof is based on the application of a $q$-analogue of the arithmetic method developed by Chudnovsky, Rukhadze, and Hata and of the transformation group for hypergeometric series–the group-structure approach introduced by Rhin and Viola. 
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     author = {W. V. Zudilin},
     title = {On the irrationality measure for a~$q$-analogue of $\zeta(2)$},
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W. V. Zudilin. On the irrationality measure for a~$q$-analogue of $\zeta(2)$. Sbornik. Mathematics, Tome 193 (2002) no. 8, pp. 1151-1172. http://geodesic.mathdoc.fr/item/SM_2002_193_8_a2/