Digamma function and general Fischer series in the theory of Kempner sums

IF 0.8 4区数学 Q2 MATHEMATICS

Expositiones Mathematicae Pub Date : 2024-09-02 DOI:10.1016/j.exmath.2024.125604

引用次数: 0

Abstract

The harmonic sum of the integers which are missing $p$ given digits in a base $b$ is expressed as $b log (b) / p$ plus corrections indexed by the excluded digits and expressed as integrals involving the digamma function and a suitable measure. A number of consequences are derived, such as explicit bounds, monotony, series representations and asymptotic expansions involving the zeta values at integers, and suitable moments of the measure. In the classic Kempner case of $b = 10$ and 9 as the only excluded digit, the series representation turns out to be exactly identical with a result obtained by Fischer already in 1993. Extending this work is indeed the goal of the present contribution.

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肯普纳和理论中的迪伽马函数和一般费歇尔级数

在基数 b 中缺失 p 个给定数位的整数的谐和表示为 blog(b)/p 加上以缺失数位为索引的修正，并表示为涉及 digamma 函数和适当度量的积分。由此可以推导出许多结果，如明确的界限、单调性、涉及整数处zeta值的数列表示和渐近展开，以及量的适当矩。在 b=10 和 9 为唯一排除数字的经典坎普纳案例中，数列表示结果与费舍尔在 1993 年获得的结果完全相同。扩展这项工作正是本论文的目标。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

发文量

审稿时长

40 days

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