On Arithmetical Properties of Certain q-Series

Abstract.

By the use of second type Padè approximations combined with a q-iteration process, linear independence results are obtained for functions f _s (z) and f′ _s (z) defined by the series

$$ \text{f} _{s} (z) = \sum\limits_{{n = 0}}^{\infty} \frac{{z^{n} }}{{1 - q^{{sn + 1}} }},\left| q \right| < 1,s = 1,2,\quad\quad\quad(1) $$

at points $z = \pm q^j, 1\leq j \leq s$. The function values in turn correspond to q-analogues of some well known constants such as π, log2 and ζ(2), the Riemann zeta function at point 2.