• 대한수학회논문집
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• 제33권1호
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• pp.13-36
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• 2018

In this paper, we obtain some formulae for harmonic sums, alternating harmonic sums and Stirling number sums by using the method of integral representations of series. As applications of these formulae, we give explicit formula of several quadratic and cubic Euler sums through zeta values and linear sums. Furthermore, some relationships between harmonic numbers and Stirling numbers of the first kind are established.

• 대한수학회논문집
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• 제29권2호
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• pp.239-255
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• 2014

We develop a set of identities for Euler type sums. In particular we investigate products of shifted harmonic numbers of order two and reciprocal binomial coefficients.

• 충청수학회지
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• 제26권4호
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• pp.769-780
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• 2013

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine and log-cosine integrals have been evaluated, in the existing literature on the subject, in many different ways. Very recently, Choi [6] presented explicit evaluations of some families of log-sine and log-cosine integrals by making use of the familiar Beta function. In the present sequel to the investigation [6], we evaluate the log-sine and log-cosine integrals involved in more complicated integrands than those in [6], by also using the Beta function.

• 대한수학회보
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• 제58권5호
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• pp.1079-1095
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• 2021

In this paper, we give new congruences with the generalized Catalan numbers and harmonic numbers modulo p². One of our results is as follows: for prime number p > 3, $${\sum\limits_{k=(p+1)/2}^{p-1}}\;k^2B_{p,k}B_{p,k-(p-1)/2}H_k{\equiv}(-1)^{(p-1)/2}\(-{\frac{521}{36}}p-{\frac{1}{p}}-{\frac{41}{12}}+pH^2_{3(p-1)/2}-10pq^2_p(2)+4\({\frac{10}{3}}p+1\)q_p(2)\)\;(mod\;p^2),$$ where q_p(2) is Fermat quotient.

• 대한수학회보
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• 제54권4호
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• pp.1255-1280
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• 2017

In this paper we present a new family of identities for parametric Euler sums which generalize a result of David Borwein et al. [2]. We then apply it to obtain a family of identities relating quadratic and cubic sums to linear sums and zeta values. Furthermore, we also evaluate several other series involving harmonic numbers and alternating harmonic numbers, and give explicit formulas.

• 대한수학회보
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• 제61권3호
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• pp.671-697
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• 2024

In this paper, we formally introduce the notion of a general parametric digamma function Ψ(−s; A, a) and we find the Laurent expansion of Ψ(−s; A, a) at the integers and poles. Considering the contour integrations involving Ψ(−s; A, a), we present some new identities for infinite series involving Dirichlet type parametric harmonic numbers by using the method of residue computation. Then applying these formulas obtained, we establish some explicit relations of parametric linear Euler sums and some special functions (e.g. trigonometric functions, digamma functions, Hurwitz zeta functions etc.). Moreover, some illustrative special cases as well as immediate consequences of the main results are also considered.

• 호남수학학술지
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• 제26권3호
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• pp.281-286
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• 2004

We calculate the harmonic series $E_{2p}:=^P\;_{^{\infty}_{n=1}}{\frac{1}{n^{2p}}$ via Waring's formula.

• 대한수학회보
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• 제57권2호
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• pp.489-508
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• 2020

This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polylogarithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.

• 대한수학회논문집
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• 제18권4호
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• pp.781-789
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• 2003

Formal manipulations of double series are useful in getting some other identities from given ones and evaluating certain summations, involving double series. The main object of this note is to summarize rather useful double series manipulations scattered in the literature and give their generalized formulas, for convenience and easier reference in their future use. An application of such manipulations to an evaluation for Euler sums (in itself, interesting), among other things, will also be presented to show usefulness of such manipulative techniques.

• 지구물리와물리탐사
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• 제14권2호
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• pp.133-139
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• 2011

지구조석을 계산하는데 있어 가장 주요한 항은 달이나 태양에 의한 구면조화차수가 2인 항으로서 지구조석의 약 98%를 이루며, 나머지는 3차 이상의 고차 항들이다. 그런데 차수 3항과 그 이상의 고차수의 항들의 값들은 차수 2항과는 다른 상수가 각각 곱하여져서 계산되어야 한다. 본 연구에서는 조석에 의한 중력변화, 변위 및 수직선의 변화에 대한 바른 계산을 검토하였으며, 필요한 관련상수들을 다시 구하였다. 동일차수내의 다른 계수의 조화함수적 조석주기의존성을 토의하였다. 한편, 지구조석을 계산하는데 관련된 중요한 개념들을 요약 소개하였다.