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On the irrationality measure for a $q$-analogue of $\zeta(2)$

W. V. Zudilin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: 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.

UDC: 511.3

MSC: Primary 11J72, 11J82; Secondary 33D15

Received: 08.11.2001

DOI: 10.4213/sm674

 English version: Sbornik: Mathematics, 2002, 193:8, 1151–1172

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