### Abstract

Let βε{lunate}(0,1) be irrational and let {nβ} denote the fractional part of nβ, n≥1. The uniform distribution of {nβ}, n≥1, implies that lim n⇒∞ 1 n ∑ j=1 ng({jβ})= sh{phonetic} 0 1g(t)dt, for each bounded and Riemann integrable g. Hardy and Littlewood proved that this relation persists when g has integrable singularities at 0 and 1, under suitable conditions on g and β. We show that by choosing suitable β, and g with an arbitrarily weak singularity at a suitable interior point gaε{lunate}(0,1), one can ensure that lim n⇒∞ 1 n ∑ j=1 ng({jβ})=∞. On the other hand, if the singularity lies at 0, then at least lim inf n⇒∞ 1 n ∑ j=1 ng({jβ})= sh{phonetic} 0 1g(t)dt. The motivation for these results lies in determination of the radius of convergence of the q or basic hypergeometric series f(z):=1+ ∑ j=1 ∞ { θ k=1 j (A-q^{k})}z^{j}, the solution of the functional equation f(z)(1-zA)=1-zqf(qz). Especially for |A| = |q| = 1, these power series are of interest in Padé approximation. Although the radius of convergence is 1 for "most" A and q, we show that f may be a transcendental entire function for suitable |A| = |q| = 1.

Original language | English (US) |
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Pages (from-to) | 469-481 |

Number of pages | 13 |

Journal | Indagationes Mathematicae |

Volume | 2 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1991 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Indagationes Mathematicae*,

*2*(4), 469-481. https://doi.org/10.1016/0019-3577(91)90032-3