Sharp inequalities for the harmonic numbers

CHAO-PING CHEN;

doi:10.37193/cjm.2012.02.15

Sharp inequalities for the harmonic numbers

Carpathian Journal of Mathematics ◽

10.37193/cjm.2012.02.15 ◽

2012 ◽

Vol 28 (2) ◽

pp. 223-229

Author(s):

CHAO-PING CHEN ◽

Keyword(s):

Harmonic Number ◽

Harmonic Numbers ◽

Double Inequality

Let Hn be the nth harmonic number, and let γ be the Euler-Mascheroni constant. We prove that for all integers n ≥ 1, the double-inequality ... holds with the best possible constants ... We also establish inequality for the Euler-Mascheroni constant.

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Some properties of harmonic numbers

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/012.2020.57.2.1458 ◽

2020 ◽

Vol 57 (2) ◽

pp. 207-216

Author(s):

Bing-Ling Wu ◽

Xiao-Hui Yan

Keyword(s):

Harmonic Number ◽

Harmonic Numbers ◽

Logarithmic Density

AbstractLet Hn be the n-th harmonic number and let vn be its denominator. It is known that vn is even for every integer . In this paper, we study the properties of Hn and prove that for any integer n, vn = en(1+o(1)). In addition, we obtain some results of the logarithmic density of harmonic numbers.

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An Extension of a Congruence by Tauraso

ISRN Combinatorics ◽

10.1155/2013/363724 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Romeo Meštrović

Keyword(s):

Positive Integer ◽

Elementary Proof ◽

Harmonic Number ◽

Combinatorial Identity ◽

Harmonic Numbers

For a positive integer let be the th harmonic number. In this paper we prove that, for any prime , . Notice that the first part of this congruence is proposed in 2008 by Tauraso. In our elementary proof of the second part of the above congruence we use certain classical congruences modulo a prime and the square of a prime, some congruences involving harmonic numbers, and a combinatorial identity due to Hernández. Our auxiliary results contain many interesting combinatorial congruences involving harmonic numbers.

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A Double Inequality for the Harmonic Number in Terms of the Hyperbolic Cosine

Turkish Journal of Analysis and Number Theory ◽

10.12691/tjant-2-6-6 ◽

2014 ◽

Vol 2 (6) ◽

pp. 223-225 ◽

Cited By ~ 2

Author(s):

Da-Wei Niu ◽

Yue-Jin Zhang ◽

Feng Qi

Keyword(s):

Harmonic Number ◽

Double Inequality ◽

Hyperbolic Cosine

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Polynomials related to harmonic numbers and evaluation of harmonic number series II

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm110615015d ◽

2011 ◽

Vol 5 (2) ◽

pp. 212-229 ◽

Cited By ~ 10

Author(s):

Ayhan Dil ◽

Veli Kurt

Keyword(s):

Subject Classification ◽

Harmonic Number ◽

Harmonic Numbers ◽

Number Series ◽

Exponential Polynomials ◽

2000 Mathematics Subject Classification

In this paper we focus on r-geometric polynomials, r-exponential polynomials and their harmonic versions. It is shown that harmonic versions of these polynomials and their generalizations are useful to obtain closed forms of some series related to harmonic numbers. 2000 Mathematics Subject Classification. 11B73, 11B75, 11B83.

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Second order alternating harmonic number sums

Filomat ◽

10.2298/fil1613511s ◽

2016 ◽

Vol 30 (13) ◽

pp. 3511-3524 ◽

Cited By ~ 2

Author(s):

Anthony Sofo

Keyword(s):

Closed Form ◽

Second Order ◽

Integral Representations ◽

Harmonic Number ◽

Binomial Coefficients ◽

Harmonic Numbers

We develop new closed form representations of sums of alternating harmonic numbers of order two and reciprocal binomial coefficients. Moreover we develop new integral representations in terms of harmonic numbers of order two.

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PROOF OF A CONGRUENCE FOR HARMONIC NUMBERS CONJECTURED BY Z.-W. SUN

International Journal of Number Theory ◽

10.1142/s1793042112500649 ◽

2012 ◽

Vol 08 (04) ◽

pp. 1081-1085 ◽

Cited By ~ 4

Author(s):

ROMEO MEŠTROVIĆ

Keyword(s):

Positive Integer ◽

Harmonic Number ◽

Harmonic Numbers ◽

Modulo P

For a positive integer n let [Formula: see text] be the nth harmonic number. In this paper, we prove that for any prime p ≥ 7, [Formula: see text] which confirms the conjecture recently proposed by Z.-W. Sun. Furthermore, we also prove two similar congruences modulo p2.

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Summations of Inverse Generalized Harmonic Numbers with Riordan Arrays

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.687-691.1394 ◽

2014 ◽

Vol 687-691 ◽

pp. 1394-1398

Author(s):

Gao Wen Xi ◽

Zheng Ping Zhang

Keyword(s):

Stirling Numbers ◽

Harmonic Number ◽

Harmonic Numbers ◽

Riordan Arrays ◽

Combinatorial Sums ◽

Generalized Harmonic Numbers ◽

Riordan Array

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we using connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. we proved some combinatorial sums and inverse generalized harmonic number identities.

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Inverse Generalized Harmonic Numbers with Riordan Arrays

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.842.750 ◽

2013 ◽

Vol 842 ◽

pp. 750-753

Author(s):

Gao Wen Xi ◽

Lan Long ◽

Xue Quan Tian ◽

Zhao Hui Chen

Keyword(s):

Stirling Numbers ◽

Harmonic Number ◽

Harmonic Numbers ◽

Riordan Arrays ◽

Combinatorial Sums ◽

Generalized Harmonic Numbers ◽

Riordan Array

In this paper, By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. Further, we proved some combinatorial sums and inverse generalized harmonic number identities.

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