Results 21 to 30 of about 10,640,447 (355)
The sum of the telescoping series formed by reciprocals of the cubic polynomials with three different negative integer roots [PDF]
Mathematics in Education, Research and Applications, 2020 This paper deals with the sum of a special telescoping series and is a free follow-up to author’s preceding paper. The terms of this series are reciprocals of the cubic polynomial with three different negative integer roots.
Radovan Potůček
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Summation Formulas Involving Binomial Coefficients, Harmonic Numbers, and Generalized Harmonic Numbers [PDF]
Abstract and Applied Analysis, 2014 A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics, and theoretical physics.
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Odd harmonic numbers exceed 10²⁴ [PDF]
Mathematics of Computation, 2010 A numbern>1n>1is harmonic ifσ(n)∣nτ(n)\sigma (n)\mid n\tau (n), whereτ(n)\tau (n)andσ(n)\sigma (n)are the number of positive divisors ofnnand their sum, respectively. It is known that there are no odd harmonic numbers up to101510^{15}. We show here that, for any odd numbern>106n>10^6,τ(n)≤n1/3\tau (n)\le n^{1/3}.
Cohen, Graeme L., Sorli, Ronald M.
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Algebraic Relations Between Harmonic Sums and Associated Quantities [PDF]
, 2003 We derive the algebraic relations of alternating and non-alternating finite harmonic sums up to the sums of depth~6. All relations for the sums up to weight~6 are given in explicit form.
Anastasiou +156 more
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Families of Integrals of Polylogarithmic Functions
Mathematics, 2019 We give an overview of the representation and many connections between integrals of products of polylogarithmic functions and Euler sums. We shall consider polylogarithmic functions with linear, quadratic, and trigonometric arguments, thereby producing ...
Anthony Sofo
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On the denominators of harmonic numbers, III [PDF]
Periodica Mathematica Hungarica, 2022 Let ℒ be the set of all positive integers n such that the denominator of 1+1/2+⋯+1/n is less than the least common multiple of 1,2,⋯,n. In this paper, under a certain assumption on linear independence, we prove that the set ℒ has the upper asymptotic density 1. The assumption follows from Schanuel’s conjecture.
Wu, Bing-Ling, Yan, Xiao-Hui
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Median Bernoulli Numbers and Ramanujan’s Harmonic Number Expansion
Mathematics, 2022 Ramanujan-type harmonic number expansion was given by many authors. Some of the most well-known are: Hn∼γ+logn−∑k=1∞Bkk·nk, where Bk is the Bernoulli numbers.
Kwang-Wu Chen
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Harmonic numbers, harmonic series and zeta function [PDF]
Moroccan Journal of Pure and Applied Analysis, 2018 AbstractThis paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the seriesis(z)=∑n=1∞μ(n)nszn$\mathcal{M}{i_s}(z) = \sum\limits_{
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Some identities involving harmonic numbers [PDF]
Mathematics of Computation, 1990 Let H n {H_n} denote the nth harmonic number. Explicit formulas for sums of the form ∑ a k H k \sum {a_k}{H_k} or ∑ a
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Non-Uniform Time Sampling for Multiple-Frequency Harmonic Balance Computations [PDF]
, 2013 A time-domain harmonic balance method for the analysis of almost-periodic (multi-harmonics) flows is presented. This method relies on Fourier analysis to derive an efficient alternative to classical time marching schemes for such flows.
Dufour, Guillaume +5 more
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