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.

Arithmetical functions associated with divisibility sequences
Original research paper. Pages 328–334
Anthony G. Shannon
Full paper (PDF, 805 Kb) | Abstract

This note looks at some aspects of divisibility sequences and generalized integers, including so-called Fermatian numbers, and extensions of ideas of Mollie Horadam and Morgan Ward.

A proof of Spence’s formula using the reciprocity law for Dedekind sums
Original research paper. Pages 335–341
Steven Brown
Full paper (PDF, 218 Kb) | Abstract

In 1963, Edward Spence published a proof of the following:

With $\phi$ being Euler’s totient function, if $n>1$ is an integer, and if

$\begin{equation*} 0<a_1<\cdots<a_{\phi(n)}<n, \end{equation*}$

are the positive integers less than $n$ , coprime with $n$ , then

$\begin{equation*} \sum_{j=1}^{\phi(n)}ja_j = \frac{\phi(n)}{24}\left(8n\phi(n)+6n+2\phi(m)(-1)^{\omega(m)}-2^{\omega(m)}\right), \end{equation*}$

where $m$ is the square-free part of $n$ , and $\omega(m)$ is the number of prime factors of $m$ .

Spence’s proof relies on an ingenious observation considering Nagell’s totient function.
Later in 1971, Lucien Van Hamme provided an alternative proof of the result using Fourier analysis and previous work from Hubert Delange in 1968. In this paper, I propose another proof of the formula using the reciprocity law for Dedekind sums. If the formula is of interest on its own, it also plays a role in the analysis of the distribution of the $a_j$ as suggested by the work from Hubert Delange.

Some new inequalities for the q-gamma and related functions, II
Original research paper. Pages 342–353
József Sándor
Full paper (PDF, 274 Kb) | Abstract

As a continuation of [5], we offer new inequalities for Jackson’s $q$ -gamma function $\Gamma_q(x).$ For example, we obtain a $q$ -analogue of the famous Jordan inequality for $(\sin x)/x$ for $x\in (0, \pi/2).$ Related inequalities, and other relations, such as the limit relations for the $q$ -gamma constant $\gamma_q,$ are also pointed out.

On (k,t)-Fibonacci–François numbers
Original research paper. Pages 354–373
Élis G. C. Mesquita, Francisco R. V. Alves, Eudes A. Costa
Full paper (PDF, 284 Kb) | Abstract

In this article, we present a study of a new member of the family of $k$ -Fibonacci numbers, which we call the $(k,t)$ -Fibonacci–François sequence $\{w_{n}^{(k,t)}\}_{ {n\ge0}}$ . This sequence is defined by the same $k$ -Fibonacci recurrence relation with the initial terms $w_{0}^{(k,t)} = k-2$ and $w_{1}^{(k,t)} = k^2 - k + 2 + t$ . We describe the structure of this family of sequences by providing explicit formulas and establishing several related algebraic identities. In addition, we derive a Binet-type formula, and extend several classical identities, including those of Tagiuri–Vajda and d’Ocagne, as well as some expressions for the negative indices. Furthermore, we investigate fundamental properties of this family, obtaining limit identities for the ratios of successive terms, as well as summation formulas for the partial sums of the $(k, t)$ -Fibonacci–François sequence.

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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