• Journal of applied mathematics & informatics
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• 제41권6호
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• pp.1227-1235
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• 2023

The study of convolution sums for divisor functions is an area that has been extensively researched by many mathematicians including Ramanujan. The aim of this paper is to find the formula for convolution sum of divisor functions with coprime conditions.

• 호남수학학술지
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• 제37권2호
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• pp.149-185
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• 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
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• 제36권5_6호
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• pp.435-446
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• 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.

• 대한수학회보
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• 제57권2호
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• pp.345-354
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• 2020

In this paper, we first give some representations for four new mock theta functions defined by Andrews [1] and Bringmann, Hikami and Lovejoy [5] using divisor sums. Then, some transformation and summation formulae for these functions and corresponding bilateral series are derived as special cases of ₂𝜓₂ series $${\sum\limits_{n=-{{\infty}}}^{{\infty}}}{\frac{(a,c;q)_n}{(b,d;q)_n}}z^n$$ and Ramanujan's sum $${\sum\limits_{n=-{{\infty}}}^{{\infty}}}{\frac{(a;q)_n}{(b;q)_n}}z^n$$.

• Journal of applied mathematics & informatics
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• 제39권3_4호
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• pp.359-380
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• 2021

The main purpose and motivation of this work is to investigate and provide some new results for coefficients derived from eta quotients related to 3. The result of this paper involve some restricted divisor numbers and their convolution sums. Also, our results give relation between the coefficients derived from infinite product, infinite sum and the convolution sum of restricted divisor functions.

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

• 호남수학학술지
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• 제36권3호
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• pp.647-657
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• 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
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• 제36권2호
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• pp.305-338
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• 2014

We investigate the explicit evaluation for the sum $\sum_{(a,b,x,y){\in}\mathbb{N}^4,\\{ax+by=n},\\{C(x,y)}$ ab in terms of various divisor functions (where C(x, y) is the set of residue conditions on x and y) for various fixed C(x, y). We also obtain some identities and congruences as interesting applications.

• 대한수학회보
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• 제49권4호
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• pp.693-704
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• 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.

• Journal of applied mathematics & informatics
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• 제41권3호
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• pp.553-567
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• 2023

There are real life situations in our lives where the things are changing continuously or from time to time. It is a very important problem for one whether to continue the existing relationship or to form a new one after some occasions. That is, people, companies, cities, countries, etc. may change their opinion or position rapidly. In this work, we think of the problem of changing relationships from a mathematical point of view and think of an answer. In some sense, we comment these changes as power changes. Our number theoretical model will be based on this idea. Using the convolution sum of the restricted divisor function E, we obtain the answer to this problem.