与某些椭圆调和级数有关的测度及Allouche-Hu-Morin极限定理

IF 0.6 3区 数学 Q3 MATHEMATICS
J.-F. Burnol
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引用次数: 0

摘要

我们考虑在长度为\(p\)的给定块(\(b\) -任意位)中出现\(k\)次的整数上的调和级数\(S(k)=\sum^{(k)} m^{-1}\),并将它们与区间[0,1)上的某些测度联系起来。我们证明了这些测度弱收敛于\(b^p\)乘以勒贝格测度,这一事实允许对Allouche, Hu和Morin[4]的定理进行新的证明,即\(\lim S(k)=b^p\log(b)\)。将给出一个定量的误差估计。组合方面涉及在Goulden-Jackson聚类生成函数形式主义范围内生成序列,以及Guibas-Odlyzko关于字符串重叠的工作。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Measures associated with certain ellipsephic harmonic series and the Allouche–Hu–Morin limit theorem

We consider the harmonic series \(S(k)=\sum^{(k)} m^{-1}\) over the integers having \(k\) occurrences of a given block of \(b\)-ary digits, of length \(p\), and relate them to certain measures on the interval [0, 1). We show that these measures converge weakly to \(b^p\) times the Lebesgue measure, a fact which allows a new proof of the theorem of Allouche, Hu, and Morin [4] which says \(\lim S(k)=b^p\log(b)\). A quantitative error estimate will be given. Combinatorial aspects involve generating series which fall under the scope of the Goulden–Jackson cluster generating function formalism and the work of Guibas–Odlyzko on string overlaps.

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来源期刊
CiteScore
1.50
自引率
11.10%
发文量
77
审稿时长
4-8 weeks
期刊介绍: Acta Mathematica Hungarica is devoted to publishing research articles of top quality in all areas of pure and applied mathematics as well as in theoretical computer science. The journal is published yearly in three volumes (two issues per volume, in total 6 issues) in both print and electronic formats. Acta Mathematica Hungarica (formerly Acta Mathematica Academiae Scientiarum Hungaricae) was founded in 1950 by the Hungarian Academy of Sciences.
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