On the radius of convergence of \(q\)-series
type: Item
instance of: scholarly article
title: On the radius of convergence of \(q\)-series
zbMATH Open document ID: 0787.40003
DOI: 10.1016/0019-3577(92)90041-I
author: V. Pereyra
published in: Indagationes Mathematicae. New Series
publication date: 1993-12-06T00:00:00Z
review text: 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 \(f(x)=\sum^ \infty_{k=0} (q_ k (x) x-p_ k(x))\).
Mathematics Subject Classification ID: 40A05
Mathematics Subject Classification ID: 11J72
Mathematics Subject Classification ID: 11L03
Mathematics Subject Classification ID: 40A15
zbMATH DE Number: 469242
zbMATH Keywords: irrationality
zbMATH Keywords: radius of convergence
zbMATH Keywords: \(q\)-series
zbMATH Keywords: continued fractions
reviewed by: Jaroslav Hančl
MaRDI profile type: Publication
cites work: Convergence of Padé approximants for a \(q\)-hypergeometric series (Wynn's power series I)
cites work: Irregular distribution of \(\{n\beta{}\} \), \(n=1,2,3,\)\dots , quadrature of singular integrands, and curious basic hypergeometric series
cites work: Q4040884
cites work: Q5835375
cites work: Q3232347
description: scientific article; zbMATH DE number 469242