## Rational approximations to a $$q$$-analogue of $$\pi$$ and some other $$q$$-series.(English)Zbl 1213.11146

Schlickewei, Hans Peter (ed.) et al., Diophantine approximation. Festschrift for Wolfgang Schmidt. Based on lectures given at a conference at the Erwin Schrödinger Institute, Vienna, Austria, 2003. Wien: Springer (ISBN 978-3-211-74279-2/hbk). Developments in Mathematics 16, 123-139 (2008).
The paper deals with the irrationality and the upper bound for the irrationality exponent of the sum of the $$q$$-series $$x \sum_{n=1}^\infty \frac{z^n}{p^n-x}$$ where $$q=p^{-1}$$, $$p\in \mathbb Z \setminus \{ 0,1,-1\}$$, $$x\in\mathbb Q$$, $$z\in \mathbb Q$$ and under the other certain conditions for $$p$$, $$x$$ and $$z$$. As an application the authors derive the upper bound for the irrationality exponent of $$\pi_q$$ which is the $$q$$-analog of $$\pi$$. The proofs make use several properties of hypergeometric and integral constructions.
For the entire collection see [Zbl 1143.11004].

### MSC:

 11J72 Irrationality; linear independence over a field 11J82 Measures of irrationality and of transcendence 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$

### Keywords:

$$q$$-series; irrationality; $$q$$-analog of $$\pi$$