TY - JOUR

T1 - Irregular distribution of {nβ}, n=1,2,3,..., quadrature of singular integrands, and curious basic hypergeometric series

AU - Driver, K. A.

AU - Lubinsky, D. S.

AU - Petruska, G.

AU - Sarnak, P.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1991

Y1 - 1991

N2 - 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-qk)}zj, 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.

AB - 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-qk)}zj, 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.

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U2 - 10.1016/0019-3577(91)90032-3

DO - 10.1016/0019-3577(91)90032-3

M3 - Article

AN - SCOPUS:0039930999

SN - 0019-3577

VL - 2

SP - 469

EP - 481

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

IS - 4

ER -