Summing the “exactly one 42” and similar subsums of the harmonic series

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics Pub Date : 2024-09-20 DOI:10.1016/j.aam.2024.102791

{"title":"Summing the “exactly one 42” and similar subsums of the harmonic series","authors":"","doi":"10.1016/j.aam.2024.102791","DOIUrl":null,"url":null,"abstract":"<div>For <math><mi>b</mi><mo>></mo><mn>1</mn></math> and αβ a string of two digits in base b, let <math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math> be the subsum of the harmonic series with only those integers having exactly one occurrence of αβ. We obtain a theoretical representation of such <math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math> series which, say for <math><mi>b</mi><mo>=</mo><mn>10</mn></math>, allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the <math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math> sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation.</div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824001234/pdfft?md5=2a1220cc0cdb8447beb302719d095400&pid=1-s2.0-S0196885824001234-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0196885824001234","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

For $b > 1$ and αβ a string of two digits in base b, let $K_{1}$ be the subsum of the harmonic series with only those integers having exactly one occurrence of αβ. We obtain a theoretical representation of such $K_{1}$ series which, say for $b = 10$ , allows computing them all to thousands of digits. This is based on certain specific measures on the unit interval and the use of their Stieltjes transforms at negative integers. Integral identities of a combinatorial nature both explain the relation to the $K_{1}$ sums and lead to recurrence formulas for the measure moments allowing in the end the straightforward numerical implementation.

查看原文本刊更多论文

求谐波数列的 "恰好一个 42 "和类似子和

对于 b>1，αβ 是一个以 b 为底数的两位数字符串，让 K1 成为谐数列的子集，其中只包含那些αβ 恰好出现一次的整数。我们可以从理论上表示这样的 K1 数列，比如对于 b=10 的数列，可以将它们计算到数千位。这是基于单位区间上的某些特定度量，以及在负整数处使用它们的斯蒂尔杰斯变换。组合性质的积分等式既解释了与 K1 和的关系，又引出了度量矩的递推公式，最终可以直接用数字实现。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Advances in Applied Mathematics 数学-应用数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

85 days

期刊介绍： Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.