Some identities on generalized harmonic numbers and generalized harmonic functions

IF 2 3区数学 Q1 MATHEMATICS

Demonstratio Mathematica Pub Date : 2022-12-30 DOI:10.1515/dema-2022-0229

Dae San Kim, H. Kim, Taekyun Kim

引用次数: 2

Abstract

Abstract The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) {F}_{n}\left(x)=B\left(x+1,n+1),\left(n=0,1,2,\ldots ) using elementary methods. For instance, we show that the Hurwitz zeta function ζ ( x + 1 , r ) \zeta \left(x+1,r) and r ! r\! are expressed in terms of those numbers and functions, for every r = 2 , 3 , 4 , 5 r=2,3,4,5 .

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广义调和数与广义调和函数的若干恒等式

调和数和广义调和数经常出现在许多不同的领域，如组合问题、解析数论中许多涉及特殊函数的表达式、算法分析等。本文的目的是利用初等方法，从函数F n (x)=B (x+1,n+1)， (n=0,1,2，…){F＿n}{}\left (x)=B \left (x+1,n+1)， \left (n=0,1,2, \ldots)中导出一些涉及广义调和数和广义调和函数的恒等式。例如，我们证明了Hurwitz ζ函数ζ (x+1,r) \zeta\left (x+1,r)和r !r\!用这些数和函数表示，对于每个r=2,3,4,5 r=2,3,4,5。

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

Demonstratio Mathematica MATHEMATICS-

CiteScore

2.40

自引率

5.00%

发文量

审稿时长

35 weeks

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