On the radius of convergence of \(q\)-series
The main result of this paper is that for every \(1<R<\infty\) there is a pair of irrationals \(\alpha,\beta\), such that the \(q\)-series \[ F(z)=1+ \sum^ \infty_{j=1} \left( \prod^ j_ {k=1} (e^{2\pi i \alpha}- e^{2 \pi ik \beta}) \right)z^ j \] has the radius of the convergence equal to \(R\). The similar result for \(\alpha\) rational and \(\beta\) irrational is included, too. Proofs are based on the approximation of continued fractions. In addition if certain conditions hold, the author proves the linear independence of \(1,x\) and \(f(x)=\sum^ \infty_{k=0} (q_ k (x) x-p_ k(x))\).
scientific article; zbMATH DE number 469242