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].


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