Volume 31, 2025, Number 2 (Online First)

Volume opened: 1 May 2025
Status: In progress

The quaternion-type cyclic-balancing sequence in groups
Original research paper. Pages 201–210
Nazmiye Yılmaz, Esra Kırmızı Çetinalp and Ömür Deveci
Full paper (PDF, 232 Kb) | Abstract

In this research, several types of definitions of the quaternion-type cyclic-balancing sequence are presented. The Cassini formula and generating function of these sequences are also obtained for all types. The quaternion-type cyclic-balancing sequences modulo $m,$ the first step to transferring this topic to group theory, are examined. These sequences in finite groups are then defined. Eventually, the lengths of periods for these sequences of the generalized quaternion group are calculated.

A new approach to tridiagonal matrices related to the Sylvester–Kac matrix
Original research paper. Pages 211–227
Efruz Özlem Mersin and Mustafa Bahşi
Full paper (PDF, 307 Kb) | Abstract

The Sylvester–Kac matrix, a well-known tridiagonal matrix, has been extensively studied for over a century, with various generalizations explored in the literature. This paper introduces a new type of tridiagonal matrix, where the matrix entries are defined by an integer sequence, setting it apart from the classical Sylvester–Kac matrix. The aim of this paper is to investigate several fundamental properties of this generalized matrix, including its algebraic structure, determinant, inverse, LU decomposition, characteristic polynomial, and various norms.

Alternative solutions to the Legendre’s equation $x^2+ky^2=z^2$
Original research paper. Pages 228–235
Kanwara Mukkhata and Sompong Chuysurichay
Full paper (PDF, 243 Kb) | Abstract

In this paper, we aim to provide alternative solutions of the Legendre’s equation $x^2 +ky^2 = z^2,$ where $k$ is a square-free positive integer. The results also lead to solutions of the well-known Pythagorean triples and Eisenstein triples.

Bivariate Leonardo polynomials and Riordan arrays
Original research paper. Pages 236–250
Yasemin Alp and E. Gökçen Koçer
Full paper (PDF, 234 Kb) | Abstract

In this paper, bivariate Leonardo polynomials are defined, which are closely related to bivariate Fibonacci polynomials. Bivariate Leonardo polynomials are generalizations of the Leonardo polynomials and Leonardo numbers. Some properties and identities (Cassini, Catalan, Honsberger, d’Ocagne) for the bivariate Leonardo polynomials are obtained. Then, the Riordan arrays are defined by using bivariate Leonardo polynomials.

Elementary proof of W. Schulte’s conjecture
Original research paper. Pages 251–255
Djamel Himane
Full paper (PDF, 187 Kb) | Abstract

In the On-Line Encyclopedia of Integer Sequences, we find the sequence $1, 1, 3, 18, 180, 2700, 56700, 1587600, 57153600, \dots,$ which is given by the formula $A_{n} = n!(n-1)!/2^{n-1}.$ On the same page, Werner Schulte conjectured that for all $n > 1,$ $n$ divides $2A_{n-1} + 4$ if and only if $n$ is prime. In this paper, we employ elementary methods to provide a simple proof of this conjecture.

Common values of two k-generalized Pell sequences
Original research paper. Pages 256–268
Zafer Şiar, Florian Luca and Faith Shadow Zottor
Full paper (PDF, 275 Kb) | Abstract

Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be the $k$ -generalized Pell sequence defined by

$\begin{equation*} P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+\cdots +P_{n-k}^{(k)} \end{equation*}$

for $n\geq 2$ with initial conditions

$\begin{equation*} P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdot \cdot \cdot =P_{-1}^{(k)}=P_{0}^{(k)}=0,\text{ and }P_{1}^{(k)}=1. \end{equation*}$

In this study, we look at the equation $P_{n}^{(k)}=P_{m}^{(l)}$ in positive integers $n,m,k,l$ such that $2\leq l<k$ and show that it has only trivial solution, namely $n=m.$

An algorithm for complex factorization of the bi-periodic Fibonacci and Lucas polynomials
Original research paper. Pages 269–279
Baijuan Shi and Can Kızılateş
Full paper (PDF, 225 Kb) | Abstract

In this paper, we consider the factorization of generalized sequences, by employing a method based on trigonometric identities. The new method is of reduced complexity and represents an improvement compared to existing results. We establish a connection between the bi-periodic Fibonacci and Lucas polynomials and tridiagonal matrices, which exploits the calculation of eigenvalues of associated tridiagonal matrices.

Weighted sum of the sixth powers of Horadam numbers
Original research paper. Pages 280–288
Kunle Adegoke, Chiachen Hsu and Olawanle Layeni
Full paper (PDF, 175 Kb) | Abstract

Ohtsuka and Nakamura found simple formulas for $\sum_{j=1}^n{F_j^6}$ and $\sum_{j=1}^n{L_j^6},$ where $F_k$ and $L_k$ are the $k$ -th Fibonacci and Lucas numbers, respectively. In this note we extend their results to a general second order sequence by deriving a formula for $\sum_{j=1}^n{(-1/q^3)^jw_{j + t}^6},$ where $(w_j(w_0,w_1;p,q))$ is the Horadam sequence defined by $w_0,\,w_1;\,w_j = pw_{j - 1} - qw_{j - 2}\, (j \ge 2);$ where $t$ is an arbitrary integer and $w_0,$ $w_1,$ $p$ and $q$ are arbitrary complex numbers, with $p\ne 0$ and $q\ne 0.$ As a by-product we establish a divisibility property for the generalized Fibonacci sequence.

Identical equations for multiplicative functions
Original research paper. Pages 289–298
Pentti Haukkanen
Full paper (PDF, 210 Kb) | Abstract

We examine identical equations for multiplicative functions and certain special cases, such as totients and quadratics. We confine ourselves to identical equations expressing the value $f(mn)$ (or the value $f(m)f(n)$ ) nontrivially in terms of the values $f(m/a)f(n/b)$ and $f(mn/(ab)),$ where $a\mid m$ and $b\mid n,$ and holding for all $m$ and $n.$ Particular attention is paid to Busche–Ramanujan type identities. We characterize all functions that satisfy the identical equations. Quasi-multiplicative functions are central to this discussion.

This volume of the International Journal “Notes on Number Theory and Discrete Mathematics” is published with the financial support of the Bulgarian National Science Fund, Grant Ref. No. KP-06-NP6/12/02.12.2024.

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