Volume 32, 2026, Number 2 (Online First)

Volume 32 ▶ Number 1 ▷ Number 2 (Online First)

Volume opened: 1 April 2026
Status: In progress

On the inverse of a certain triangular matrix and its connection to the largest odd divisor
Original research paper. Pages 255–262
Sela Fried
Full paper (PDF, 196 Kb) | Abstract

The inverse $R$ of a certain infinite triangular matrix $A$ is shown to be directly related to the largest odd divisor function, thus proving a conjecture of Barry. We also provide a proof of a formula for $R$ given by Yin and obtain bivariate generating functions for $A$ and $R$ .

A note on Leibniz rule for difference quotient
Original research paper. Pages 263–268
Taekyun Kim and Dae San Kim
Full paper (PDF, 161 Kb) | Abstract

In this note, we derive a Leibniz rule for difference quotient.

Simpler congruences for Jacobi sum J(1,1)₄₉ of order 49
Original research paper. Pages 269–280
Ishrat Jahan Ansari, Vikas Jadhav and Devendra Shirolkar
Full paper (PDF, 237 Kb) | Abstract

Congruences of order $l^2$ (with $l$ an odd prime) were obtained by D. Shirolkar and S. A. Katre [15]. In this paper we determine congruence of Jacobi sums $J(1,1)_{49}$ of order $49$ over a field ${\mathbb F}_{p}$ . We also show that simpler congruences hold for $J(1,1)_{49}$ in the case of artiad and hyperartiad primes.

q-Leonardo polynomials
Original research paper. Pages 281–299
Naim Tuglu, Sude Sıla Ekinci and Miraç Çetin Keskin
Full paper (PDF, 327 Kb) | Abstract

In this study, we define the $q$ -Leonardo Pisano polynomials and $q$ -Leonardo Lucas polynomials of the first and the second kinds, collectively termed $q$ -Leonardo polynomials. We investigate some interesting properties of the new polynomials.

Generalizing Pascal’s row sums to complex orders: A Poisson summation approach
Original research paper. Pages 300–312
Fabrizio Mancini
Full paper (PDF, 808 Kb) | Abstract

We generalize Pascal’s triangle row sums by defining shifted fractional binomial coefficients with a real scaling parameter and a complex shift. Using the Poisson summation formula, we derive exact summation identities extended via analytic continuation to conditionally convergent regimes. We prove that when the magnitude of the scaling parameter is not greater than 2, the sum collapses to a simple exponential form independent of the shift; otherwise, it generalizes classical series multisections to non-integer steps and complex shifts. These closed forms allow for rapid and exact computation even where symbolic algebra systems fail.

The homothetical-Hopf transformation of Pythagorean quadruples
Original research paper. Pages 313–320
Mircea Crasmareanu
Full paper (PDF, 226 Kb) | Abstract

This note introduces a transformation of Pythagorean quadruples by using the composition between the Hopf map and a homothety of the space ℝ³. Both the real algebra of complex numbers and the algebra of quaternions are used in this construction. Three examples are detailed, the first one concerning the well-known twin Pythagorean quadruple (1, 2, 2, 3). The trigonometric parametrization of the Euclidean unit sphere S² ⊂ 𝔼³ allows us to prove that this transformation does not produce twin Pythagorean quadruples. A matrix approach for our transformation is also presented.

A note on h-fold signed sumset in the set of integers
Original research paper. Pages 321–327
Mohan
Full paper (PDF, 176 Kb) | Abstract

Let $h$ and $k$ be positive integers, and let $A = \{ a_{0}, a_{1},\ldots, a_{k-1}\}$ be a finite set of $k$ integers. The h-fold signed sumset, denoted by $h_{\pm}A$ , is defined as

$\begin{align*} h_{\pm}A := \left\lbrace \sum_{i=0}^{k-1} \lambda_{i} a_{i}: \lambda_{i} \right. \in \left\lbrace 0, \pm 1, \pm 2, \ldots, \pm h \right\rbrace \text{ for } \ i= 0&, 1, \ldots, k-1 \\ &\left. \text{ and } \sum_{i=0}^{k-1} \left| \lambda_{i} \right| = h\right\rbrace \end{align*}$

Bhanja and Pandey [J. Number Theory, 196 (2019), 340–352] gave an optimal lower bound for the cardinality of $h_{\pm}A$ . They also characterized the set $A$ when the cardinality of $h_{\pm}A$ attains the optimal lower bound. In this note, we revisit their results by providing new proofs. We observe that the study of obtaining the optimal lower bound for the cardinality of $h_{\pm}A$ , and the structure of the set $A$ when $h_{\pm}A$ attains the optimal lower bound, rather than for an arbitrary set of integers, suffices when $A$ is an arithmetic progression.

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-NP7/2/03.12.2025. Bulgarian National Science Fund bears no liability for the content of the published materials.

Volume 32 ▶ Number 1 ▷ Number 2 (Online First)

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