• Kyungpook Mathematical Journal
• /
• 제53권4호
• /
• pp.553-563
• /
• 2013

We derive a family of non-linear differential equations from the generating functions of the Euler polynomials and study the solutions of these differential equations. Then we give some new and interesting identities and formulas for the Euler polynomials of higher order by using our non-linear differential equations.

• Kyungpook Mathematical Journal
• /
• 제61권3호
• /
• pp.473-486
• /
• 2021

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.

• 대한토목학회논문집
• /
• 제6권1호
• /
• pp.61-68
• /
• 1986

본(本) 연구(硏究)에서는 포텐셜에너지 이론(理論)을 바탕으로 탄성정력학적(彈性靜力學的) 문제(問題)에 대한 재차 Euler 방정식(方程式)을 완벽히 만족(滿足)하는 새로운 형태(形態)의 형상함수(形狀函數)를 제안(提案)하였다. Shear locking 현상(現像)을 피하기 위한 함차적분(咸差積分)이 필요(必要)없는 이 형상함수(形狀函數)를 사용(使用)한 보 유한요소(有限要素)를 Galerkin 가중잔차법(加重殘差法)으로 유도(誘導)하여 보의 자유진동(自由振動)과 정력학적(靜力學的) 문제(問題)를 해석(解析)하여 이 형상함수(形狀函數)의 효율성(効率性)과 정확도(正確度)를 고찰(考察)하였다. 본(本) 연구(硏究)에서 제안(提案)된 형상함수(形狀函數)를 이용(利用)한 유한요소해(有限要素解)는 보의 靜的解析의 경우 정확해(正確解)와 완벽히 일치(一致)하였으며 자유진동해석(自由振動解析)의 경우에도 월등(越等)한 정확도(正確度)를 보여주었다. 따라서 본(本) 연구(硏究)에서 제안(提案)된 형상함수(形狀函數)의 개념(槪念)은 모든 탄성정력학적(彈性靜力學的) 문제(問題) 유한요소해(有限要素解)를 구하는데 적용(適用)될 수 있다고 사료된다.

• 한국수학사학회지
• /
• 제20권4호
• /
• pp.71-84
• /
• 2007

베르누이가 처음으로 자연수 k에 대하여 합 $S_n(k)=\sum_{{\iota}=1}^n\;{\iota}^k$에 관한 공식들을 유도하는 방법을 발견하였다([4]). 그 이후, 리만 제타함수와 관련된 베르누이 수와 오일러 수에 관한 성질들이 연구되어왔다. 최근에 김태균은 $\mathbb{Z}_p$상에서 p-진 q-적분과 관련된 확장된 q-베르누이 수와 q-오일러 수, 연속된 q-정수의 멱수의 합에 관한 성질들을 밝혔다. 본 논문에서는 연속된 q-정수의 멱수의 합에 관한 역사적 배경과 발달과정을 고찰하고, 오일러 및 베르누이 수와 관련된 리만 제타함수가 해석적 함수로써 값을 가지는 문제를 q-확장된 부분의 이론으로 연구되어온 q-오일러 제타함수에 대해 체계적으로 논의한다.

• 호남수학학술지
• /
• 제36권3호
• /
• pp.647-657
• /
• 2014

In this paper, we study combinatoric convolution sums of divisor functions and get values of this sum when $n=2^mp$. We find that the value of this convolution sum is represented by a sum of powers of 2 and Bernoulli or Euler number.

• 호남수학학술지
• /
• 제40권4호
• /
• pp.691-700
• /
• 2018

The main object of the present research paper is to establish two (potentially) useful Euler type integrals involving multiindex Mittag-Leffler functions, which are expressed in terms of Wright hypergeometric functions. Some deductions of the main results are also indicated.

• 한국소음진동공학회:학술대회논문집
• /
• 한국소음진동공학회 2009년도 춘계학술대회 논문집
• /
• pp.573-578
• /
• 2009

Starting from the rotating beam finite element in which the interpolating shape functions satisfies the governing static homogeneous differential equation of Euler-Bernoulli rotating beams, we derived new shape functions that satisfies the governing differential equation which contains the terms of hub radius and setting angle. The shape functions are rational functions which depend on hub radius, setting angle, rotational speed and element position. Numerical results for uniform and tapered cantilever beams with and without hub radius and setting angle are compared with the available results. It is shown that the present element offers an accurate method for solving the free vibration problems of rotating beam.

• 한국소음진동공학회논문집
• /
• 제19권6호
• /
• pp.591-598
• /
• 2009

Starting from the rotating beam finite element in which the interpolating shape functions satisfy the governing static homogeneous differential equation of Euler-Bernoulli rotating beams, we derived new shape functions that satisfy the governing differential equation which contains the terms of hub radius and setting angle. The shape functions are rational functions which depend on hub radius, setting angle, rotational speed and element position. Numerical results for uniform and tapered cantilever beams with and without hub radius and setting angle are compared with the available results. It is shown that the present element offers an accurate method for solving the free vibration problems of rotating beams.

• 대한수학회보
• /
• 제61권3호
• /
• pp.671-697
• /
• 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.

• 대한수학회보
• /
• 제61권4호
• /
• pp.1033-1051
• /
• 2024

In this paper, we define a parametric variant of generalized Euler sums and construct contour integration to give some explicit evaluations of these parametric Euler sums. In particular, we establish several explicit formulas of (Hurwitz) zeta functions, linear and quadratic parametric Euler sums. Furthermore, we also give an explicit evaluation of alternating double zeta values ${\zeta}({\bar{2j}};\,2m+1)$ in terms of a combination of alternating Riemann zeta values by using the parametric Euler sums.