Volume 30, 2024, Number 4 (Online First)

Volume 30 ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4 (Online First)

Volume opened: 1 November 2024
Status: In progress

Congratulations to our Editorial Board Members Prof. Taekyun Kim, Dr. Mladen Vassilev-Missana and Prof. Krassimir Atanassov for their anniversaries in 2024!
Editorial. Pages 663–664
Vassia Atanassova
Editorial (PDF, 583 Kb)

Algorithms for representing positive odd integers as the sum of arithmetic progressions
Original research paper. Pages 665–680
Peter J.-S. Shiue, Anthony G. Shannon, Shen C. Huang, Michael R. Schwob, Rama Venkat
Full paper (PDF, 226 Kb) | Abstract

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

Quotients of sequences under the binomial convolution
Original research paper. Pages 681–690
Pentti Haukkanen
Full paper (PDF, 221 Kb) | Abstract

This paper gives expressions for the solution $\{a(n)\}$ of the equation

$\begin{equation*} \sum_{k=0}^n{n \choose k}a(k)b(n-k)=c(n), \ n=0,1,2,\ldots, \end{equation*}$

where $b(0)\ne 0$ , that is, of the equation $a \circ b = c$ in $a$ , where $\circ$ is the binomial convolution. These expressions are classified as recursive, explicit, determinant, exponential generating function and convolutional expressions. These expressions are compared with those under the usual Cauchy convolution. Several special cases and examples of combinatorial nature are also discussed.

New congruences modulo powers of 2 for k-regular overpartition pairs
Original research paper. Pages 691–703
Riyajur Rahman and Nipen Saikia
Full paper (PDF, 230 Kb) | Abstract

Let $\overline{B}_{k}(n)$ denote the number of $k$ regular overpartition pairs where a $k$ -regular overpartition pair of $n$ is a pair of $k$ -regular overpartitions $(a,b)$ in which the sum of all the parts is $n$ . Naika and Shivasankar (2017) proved infinite families of congruences for $\overline{B}_3(n)$ and $\overline{B}_4(n)$ . In this paper, we prove infinite families of congruences modulo powers of $2$ for $\overline{B}_{3\gamma}(n)$ , $\overline{B}_{4\gamma}(n)$ and $\overline{B}_{6\gamma}(n)$ .

Revisiting some r-Fibonacci sequences and Hessenberg matrices
Original research paper. Pages 704–715
Carlos M. da Fonseca, Paulo Saraiva and Anthony G. Shannon
Full paper (PDF, 244 Kb) | Abstract

The relationship between different generalizations of Fibonacci numbers and matrices is common in the literature. However, the basic relation of such sequences with Hessenberg matrices is often not properly explored. In this work we revisit some classic results and present some applications in recent contexts.

Powers of the operator $u(z)\frac{d}{dz}$ and their connection with some combinatorial numbers
Original research paper. Pages 716–734
Ioana Petkova
Full paper (PDF, 322 Kb) | Abstract

In this paper the operator $A = u(z)\frac{d}{dz}$ is considered, where $u$ is an entire or meromorphic function in the complex plane. The expansion of $A^{k}$ ( $k\geq1$ ) with the help of the powers of the differential operator $D=\frac{d}{dz}$ is obtained, and it is shown that this expansion depends on special numbers. Connections between these numbers and known combinatorial numbers are given. Some special cases of the operator $A$ , corresponding to $u(z) = z$ , $u(z) = e^{z}$ , $u(z) = \frac{1}{z}$ , are considered.

On (k,p)-Fibonacci numbers and matrices
Original research paper. Pages 735–744
Sinan Karakaya, Halim Özdemir and Tuğba Demirkol
Full paper (PDF, 253 Kb) | Abstract

In this paper, some relations between the powers of any matrices $X$ satisfying the equation $X^k-pX^{k-1}-(p-1)X-I=\bf{0}$ and $(k,p)$ -Fibonacci numbers are established with $k\geq2$ . First, a result is obtained to find the powers of the matrices satisfying the condition above via $(k,p)$ -Fibonacci numbers. Then, new properties related to $(k,p)$ -Fibonacci numbers are given. Moreover, some relations between the sequence $\{F_{3,s}(n)\}$ and the generalized Fibonacci sequence $\{U_n(p,q)\}$ are also examined.

The complex-type Pell p-numbers
Original research paper. Pages 745–754
Yeşim Aküzüm, Hüseyin Aydın and Ömür Deveci
Full paper (PDF, 245 Kb) | Abstract

In this paper, we define the complex-type Pell p-numbers and give the generating matrix of these defined numbers. Then, we produce the combinatorial representation, the generating function, the exponential representation and the sums of the complex-type Pell p-numbers. Also, we derive the determinantal and the permanental representations of the complex-type Pell p-numbers by using certain matrices which are obtained from the generating matrix of these numbers. Finally, we obtain the Binet formula for the complex-type Pell p-number.

Partitions of numbers and the algebraic principle of Mersenne, Fermat and even perfect numbers
Original research paper. Pages 755–775
A. M. S. Ramasamy
Full paper (PDF, 305 Kb) | Abstract

Let ρ be an odd prime greater than or equal to 11. In a previous work, starting from an M-cycle in a finite field 𝔽_ρ, it has been established how the divisors of Mersenne, Fermat and Lehmer numbers arise. The converse question has been taken up in a succeeding work and starting with a factor of these numbers, a method has been provided to find an odd prime ρ and the M-cycle in 𝔽_ρ contributing the factor under consideration. Continuing the study of the two previous works, a certain type of partition of a natural number is considered in the present paper. Concerning the Mersenne, Fermat and even perfect numbers, the algebraic principle is established.

A note on Diophantine inequalities in function fields
Original research paper. Pages 776–786
Kathryn Wilson
Full paper (PDF, 249 Kb) | Abstract

We will discuss how the Bentkus–Götze–Freeman variant of the Davenport–Heilbronn circle method can be used to study $\mathbb{F}_q[t]$ solutions to inequalities of the form

$\mathrm{ord}(\lambda_1p_1^k+\cdots+\lambda_sp_s^k-\gamma)<\tau ,$

where constants $\lambda_1,\dots, \lambda_s \in\mathbb{F}_q((1/t))$ satisfy certain conditions. This result is a generalization of the work done by Spencer in [11] to count the number of solutions to inequalities of the form

$\mathrm{ord}(\lambda_1p_1^k+\cdots+\lambda_sp_s^k)<\tau.$

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