• 대한수학회지
• /
• 제52권3호
• /
• pp.537-565
• /
• 2015

In this paper, we give relationship between Bernoulli-Euler polynomials and convolution sums of divisor functions. First, we establish two explicit formulas for certain combinatoric convolution sums of divisor functions derived from Bernoulli and Euler polynomials. Second, as applications, we show five identities concerning the third and fourth-order convolution sums of divisor functions expressed by their divisor functions and linear combination of Bernoulli or Euler polynomials.

• 호남수학학술지
• /
• 제35권2호
• /
• pp.251-302
• /
• 2013

We study convolution sums of certain restricted divisor functions in detail and present explicit evaluations in terms of usual divisor functions for some specific situations.

• 호남수학학술지
• /
• 제37권2호
• /
• pp.149-185
• /
• 2015

Let ${\sigma}_s(N)={\sum}_{d{\mid}N}d^s$ denote the sum of sth powers of the positive divisors of N. In this article, we consider the convolution sums of four divisor functions.

• Journal of applied mathematics & informatics
• /
• 제34권1_2호
• /
• pp.145-156
• /
• 2016

It is known that certain convolution sums can be expressed as a combination of divisor functions and Bernoulli formula. In this article, we consider relationship between fifth-order combinatoric convolution sums of divisor functions and Bernoulli polynomials. As applications of these identities, we give a concrete interpretation in terms of the procedural modeling method.

• Kyungpook Mathematical Journal
• /
• 제58권4호
• /
• pp.599-613
• /
• 2018

We study a generalization of the classical Pentagonal Number Theorem and its applications. We derive new identities for certain infinite series, recurrence relations and convolution sums for certain restricted partitions and divisor sums. We also derive new identities for Bell polynomials.

• 호남수학학술지
• /
• 제36권1호
• /
• pp.55-66
• /
• 2014

In this paper, we study the convolution sums involving odd divisor functions, and their relations to Weierstrass ${\wp}$-functions.

• 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.

• Journal of applied mathematics & informatics
• /
• 제36권5_6호
• /
• pp.435-446
• /
• 2018

Using the theory of combinatoric convolution sums, we establish some arithmetic identities involving Liouville functions and restricted divisor functions. We also prove some relations involving restricted divisor functions and Stirling numbers of the first kind for divisor functions.

• 한국멀티미디어학회논문지
• /
• 제16권5호
• /
• pp.637-646
• /
• 2013

본 연구는 다수의 나무로 구성된 실외 지형에 적합하고 다양하고 자연스러운 나무를 모델링하기 위하여 새로운 성장 규칙(약수 함수의 합성 곱 기반)의 모델링 방법을 제안한다. 기본적으로 나무를 구성하는 가지와 잎의 효율적 관리와 자연스러운 가지 증식을 위하여 기존의 성장 볼륨기반 알고리즘을 활용한다. 이 논문의 주요 특징은 각 성장 단계에서 가지와 잎의 성장과 운명을 자연스럽게 표현하는 약수 함수 합성 곱 이론을 도입하는 것이다. 이를 기반으로 일반화된 생성 함수를 갖는 여러 약수 함수와 성장 규칙의 변형을 통해 사용자의 제어를 최소화하여 다양한 나무를 모델링하는 방법을 제안한다. 이 모델링 방법은 가지와 잎을 동시에 고려하는 특징이 있으며, 다수의 나무들로 구성된 실외 지형 구축에 보다 효과적이라는 이점이 있다. 제안한 방법을 통해 자연스럽고 다양한 나무 모델 생성과 이를 활용하여 넓은 자연 지형 구축 가능성과 다수의 나무를 구성하는 과정에서의 효율성을 실험을 통해 증명한다.

• 대한수학회보
• /
• 제49권4호
• /
• pp.693-704
• /
• 2012

We consider Weierstrass functions and divisor functions arising from $q$-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. In this note, however, we obtain a similar result by differentiating further. For any $n{\geq}1$, we have $$k{\cdot}{\tau}_{2;k,l}(n)=2n{\cdot}E_{\frac{k-l}{2}}(n;k)+l{\cdot}{\tau}_{1;k,l}(n)+2k{\cdot}{\sum_{j=1}^{n-1}}E_{\frac{k-1}{2}(j;k){\tau}_{1;k,l}(n-j)$$. Finally, we shall give a table for $E_1(N;3)$, ${\sigma}(N)$, ${\tau}_{1;3,1}(N)$ and ${\tau}_{2;3,1}(N)$ ($1{\leq}N{\leq}50$) and state simulation results for them.