**Volume 25** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**An asymptotic formula for the Chebyshev theta function**

*Original research paper. Pages 1–7*

Aditya Ghosh

Full paper (PDF, 182 Kb) | Abstract

**A note on balanced numbers**

*Original research paper. Pages 8–15*

József Sándor and Krassimir T. Atanassov

Full paper (PDF, 198 Kb) | Abstract

and

for are given. Connections with related problems and inequalities are pointed out, too.

**Certain generating functions for the quadruple hypergeometric series K_{10}**

*Original research paper. Pages 16–23*

Praveen Agarwal, Jihad A. Younis and Taekyun Kim

Full paper (PDF, 179 Kb) | Abstract

*K*

_{10}. Some special cases of the main results here are also considered.

**A family of elliptic curves of rank ≥ 5 over ℚ( m)**

*Original research paper. Pages 24–29*

Arman Shamsi Zargar and Naser Zamani

Full paper (PDF, 172 Kb) | Abstract

**Some studies on Eisenstein series and their applications**

*Original research paper. Pages 30–43*

H. C. Vidya and B. R. Srivatsa Kumar

Full paper (PDF, 219 Kb) | Abstract

*π*and convolution sums.

**Formal power series in several variables**

*Original research paper. Pages 44–57*

Pentti Haukkanen

Full paper (PDF, 219 Kb) | Abstract

*n*variables as an

*n*-way array of complex or real numbers and investigate its algebraic properties without analytic tools. We also consider the formal derivative, logarithm and exponential of a formal power series in

*n*variables. Applications to multiplicative arithmetical functions in several variables and cumulants in statistics are presented.

**Differential and difference polynomial sequences**

*Original research paper. Pages 58–65*

Veasna Kim, Vichian Laohakosol and Supawadee Prugsapitak

Full paper (PDF, 201 Kb) | Abstract

*n*positive integers which represents the values of an integer polynomial at the first n positive integers. We extend this notion to differential and difference polynomial sequences which are defined analogously by incorporating not only the polynomial values but also the values of its derivatives and/or differences at integer points. Characterizations and their algebraic structures are determined.

**Multisection of series**

*Original research paper. Pages 66–71*

A. G. Shannon

Full paper (PDF, 151 Kb) | Abstract

**Distribution of constant terms of irreducible polynomials in ℤ _{p}[x]**

*Original research paper. Pages 72–82*

Sarah C. Cobb, Michelle L. Knox, Marcos Lopez, Terry McDonald and Patrick Mitchell

Full paper (PDF, 206 Kb) | Abstract

*q*over a finite field. These formulas are derived from work done by Yucas. We show that the number of polynomials of a given constant term depends only on whether the constant term is a residue in the underlying field. We further show that as

^{k}*k*becomes large, the proportion of irreducible polynomials having each constant term is asymptotically equal.

**Determinants of Toeplitz–Hessenberg matrices with generalized Fibonacci entries**

*Original research paper. Pages 83–95*

Taras Goy and Mark Shattuck

Full paper (PDF, 209 Kb) | Abstract

*S*are given for our formulas in several particular cases, including those involving the Chebyshev polynomials.

_{n}**On the sum of three arbitrary Fibonacci and Lucas numbers**

*Original research paper. Pages 96–101*

Nurettin Irmak, Zafer Şiar and Refik Keskin

Full paper (PDF, 135 Kb) | Abstract

for and a natural number . It is shown that only the equation has a finite number of solutions. The others have infinitely many solutions.

*s*-th power of Fibonacci number of the form 2^{a} + 3^{b} + 5^{c}

*Original research paper. Pages 102–109*

Nurettin Irmak and Bo He

Full paper (PDF, 235 Kb) | Abstract

**Generalized Fibonacci and k-Pell matrix sequences: Another way of demonstrating their properties**

*Original research paper. Pages 110–122*

Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino

Full paper (PDF, 239 Kb) | Abstract

commutative matrices derived from the generalized Fibonacci matrix sequence and the

*k*-Pell matrix sequence. In the present work, through the identification of certain special matrices, we can identify other forms of demonstration and also the description of commutative matrix properties for negative indices.

**On generalized bicomplex k-Fibonacci numbers**

*Original research paper. Pages 123–133*

Tülay Yağmur

Full paper (PDF, 166 Kb) | Abstract

*k*-Fibonacci numbers. We also give the generating function and Binet’s formula for these numbers. In addition, we obtain some identities such as Honsberger, d’Ocagne’s, Catalan’s, and Cassini’s identities involving the generalized bicomplex

*k*-Fibonacci numbers.

**An application of exponential sums over the divisor function**

*Original research paper. Pages 134–142*

Tippawan Puttasontiphot

Full paper (PDF, 167 Kb) | Abstract

**On bipartite graphs and the Fibonacci numbers**

*Original research paper. Pages 143–149*

Fatih Yılmaz and Pınar Eldutar

Full paper (PDF, 157 Kb) | Abstract

**On Beck’s zero-divisor graph**

*Original research paper. Pages 150–157*

Deepa Sinha and Bableen Kaur

Full paper (PDF, 154 Kb) | Abstract

*R*with unity (1 ≠ 0), the zero-divisor graph of

*R*, denoted by

*Γ*(

*R*), is a simple graph with vertices as elements of

*R*and two distinct vertices are adjacent whenever the product of the vertices is zero. This article aims at gaining a deeper insight into the basic structural properties of zero-divisor graphs given by Beck.

*This issue 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-NP-28/2018.*