Irregular distribution of \(\{n\beta{}\} \), \(n=1,2,3,\)\dots , quadrature of singular integrands, and curious basic hypergeometric series
Let \(\beta\in]0,1[\) be an irrational number. It is well-known that the sequence \((\{n\beta\})_{n\geq 1}\) of fractional parts is uniformly distributed modulo one. Further, \[ \lim_{N\to\infty}{1\over N}\sum^ N_{n=1} f(\{n\beta\})=\int^ 1_ 0 f(t)\;dt\tag{1} \] for every Riemann-integrable function \(f\) defined on \([0,1]\), \(f(0)=f(1)\). For details, the reader is referred to the comprehensive monograph of \textit{L. Kuipers} and \textit{H. Niederreiter} [Uniform distribution of sequences (Wiley, New York, 1974; Zbl 0281.10001)]. \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Proc. Camb. Philos. Soc. 42, 85-90 (1946; Zbl 0060.157)]proved that (1) holds even for functions \(f\) with integrable singularities at 0 and 1, under suitable conditions for \(f\) and \(\beta\) (``bounded partial quotients''). The authors generalize this result. They show the following: (i) If \(\psi: ]0,1]\to[0,+\infty[\) is a decreasing function with \(\lim_{t\to 0+}\psi(t)=\infty\) and finite integral, then \[ \liminf_{N\to\infty}{1\over N}\sum^ N_{n=1} \psi(\{n\beta\})=\int^ 1_ 0\psi(t)\;dt. \] (ii) No matter how weak the singularity of \(\psi\) at 0, there exist \(\beta\) for which \(\liminf\) cannot be replaced by lim. (iii) It is essential for relation (1) that the singularity of \(\psi\) lies at an endpoint of the interval. Let \(\psi\) be as in (i). Then there exist an irrational number \(\beta\in]0,1[\) and some number \(\alpha\in]0,\beta[\) such that \[ \lim_{N\to\infty}{1\over N}\sum^ N_{n=1}\psi(|\{n\beta\}-\alpha|)=\infty. \] The authors give sufficient conditions for such pairs \((\beta,\alpha)\) in terms of the continued fraction expansion of \(\beta\). (iv) Motivated by studies of the convergence of Padé approximants for certain hypergeometric series, the authors consider the series \(F(A;q;z):=1+\sum^ \infty_{j=1}\bigl(\prod^ j_{k=1}(A-q^ k)\bigr)z^ j\), where \(A\neq q^ k\), \(k\geq 0\). Here the interesting case is \(| A|=| q|=1\) and \(q=\exp(2\pi i\beta)\). The authors obtain results for the radius of convergence of the series \(F(A;q;z)\). As an example we cite, with \(\beta\) irrational: \(R(A;q)\geq 1\); if \(A=1\), then \(R(A;q)=1\); \(\exists\) a set \({\mathcal S}_ q\) of logarithmic capacity zero, such that \(R(A;q)=1\) whenever \(A\not\in{\mathcal S}_ q\).
scientific article; zbMATH DE number 60869
scientific article