**Volume 27** ▶ Number 1 ▷ Number 2 ▷ Number 3 ▷ Number 4

**Relations between R_{α}, R_{β} and R_{m} functions related to Jacobi’s triple-product identity and the family of theta-function identities**

*Original research paper. Pages 1–11*

M. P. Chaudhary

Full paper (PDF, 201 Kb) | Abstract

*R*,

_{α}*R*and

_{β}*R*functions which are based upon a number of

_{m}*q*-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper we answer a open question of Srivastava et al [33], and established relations in terms of

*R*,

_{α}*R*and

_{β}*R*(for

_{m}*m*= 1, 2, 3), and

*q*-products identities. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities.

**The abundancy index of divisors of odd perfect numbers – Part II**

*Original research paper. Pages 12–19*

Keneth Adrian Precillas Dagal and Jose Arnaldo Bebita Dris

Full paper (PDF, 198 Kb) | Abstract

*N*=

*q*

^{k}n^{2}is an odd perfect number with special prime

*q*, and

*N*is not divisible by 3, then the inequality

*q*<

*n*holds. We then give another unconditional proof for the inequality

*q*<

*n*which is independent of the results of Brown and Starni.

**Bi-unitary multiperfect numbers, V**

*Original research paper. Pages 20–40*

Pentti Haukkanen and Varanasi Sitaramaiah

Full paper (PDF, 261 Kb) | Abstract

Let denote the sum of the bi-unitary divisors of . A positive integer is called a bi-unitary multiperfect number if for some . For we obtain the bi-unitary triperfect numbers.

Peter Hagis (1987) proved that there are no odd bi-unitary multiperfect numbers. The present paper is part V in a series of papers on even bi-unitary multiperfect numbers. In parts I, II and III we determined all bi-unitary triperfect numbers of the form , where and is odd. In parts IV(a-b) we solved partly the case . In this paper we fix the case . In fact, we show that is the only bi-unitary triperfect number of the present type.

**Inequalities for generalized divisor functions**

*Original research paper. Pages 41–48*

József Sándor

Full paper (PDF, 218 Kb) | Abstract

**On two theorems of Vassilev-Missana**

*Original research paper. Pages 49–50*

Richard P. Brent

Full paper (PDF, 134 Kb) | Abstract

**A note on prime zeta function and Riemann zeta function. Corrigendum**

*Original research paper. Pages 51–53*

Mladen Vassilev-Missana

Full paper (PDF, 141 Kb) | Abstract

**Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression**

*Original research paper. Pages 54–63*

Kunle Adegoke, Robert Frontczak and Taras Goy

Full paper (PDF, 187 Kb) | Abstract

**A formula for the number of non-negative integer solutions of a_{1}x_{1} + a_{2}x_{2} + ··· + a_{m}x_{m} = n in terms of the partial Bell polynomials**

*Original research paper. Pages 64–69*

Sumit Kumar Jha

Full paper (PDF, 187 Kb) | Abstract

**Perrin’s bivariate and complex polynomials**

*Original research paper. Pages 70–78*

Renata Passos Machado Vieira, Milena Carolina dos Santos Mangueira, Francisco Regis Vieira Alves and Paula Maria Machado Cruz Catarino

Full paper (PDF, 223 Kb) | Abstract

**Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials**

*Original research paper. Pages 79–87*

Jorma K. Merikoski

Full paper (PDF, 196 Kb) | Abstract

**On the Diophantine equations z^{2} = f(x)^{2} ± f(x)f(y) + f(y)^{2}**

*Original research paper. Pages 88–100*

Qiongzhi Tang

Full paper (PDF, 251 Kb) | Abstract

**Sums of powers of integers and hyperharmonic numbers**

*Original research paper. Pages 101–110*

José Luis Cereceda

Full paper (PDF, 191 Kb) | Abstract

*k*-th powers of the first

*n*positive integers,

*S*(

_{k}*n*), that involves the hyperharmonic numbers and the Stirling numbers of the second kind. Then, using an explicit representation for the hyperharmonic numbers, we generalize this formula to the sums of powers of an arbitrary arithmetic progression. Furthermore, we express the Bernoulli polynomials in terms of hyperharmonic polynomials and Stirling numbers of the second kind. Finally, we extend the obtained formula for

*S*(

_{k}*n*) to negative values of

*n*.

**On the connections among Fibonacci, Pell, Jacobsthal and Padovan numbers**

*Original research paper. Pages 111–128*

Ömür Deveci

Full paper (PDF, 248 Kb) | Abstract

**Generalized Lucas numbers of the form 3 × 2^{m}**

*Original research paper. Pages 129–136*

Salah Eddine Rihane, Chefiath Awero Adegbindin and Alain Togbé

Full paper (PDF, 204 Kb) | Abstract

**Some identities of generalized Tribonacci and Jacobsthal polynomials**

*Original research paper. Pages 137–147*

Abdeldjabar Hamdi and Salim Badidja

Full paper (PDF, 212 Kb) | Abstract

**The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials**

*Original research paper. Pages 148–158*

Merve Taştan, Engin Özkan and Anthony G. Shannon

Full paper (PDF, 802 Kb) | Abstract

*k*-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini’s Identities for the families and their polynomials. Finally, we suggest avenues for further research.

**More identities on Fibonacci and Lucas hybrid numbers**

*Original research paper. Pages 159–167*

Nazmiye Yilmaz

Full paper (PDF, 159 Kb) | Abstract

**A short remark on a new Fibonacci-type sequence**

*Original research paper. Pages 168–171*

Krassimir T. Atanassov

Full paper (PDF, 124 Kb) | Abstract

**Classifying Galois groups of an orthogonal family of quartic polynomials**

*Original research paper. Pages 172–190*

Pradipto Banerjee and Ranjan Bera

Full paper (PDF, 159 Kb) | Abstract

**On r-dynamic coloring of comb graphs**

*Original research paper. Pages 191–200*

K. Kalaiselvi, N. Mohanapriya and J. Vernold Vivin

Full paper (PDF, 185 Kb) | Abstract

**Refined enumeration of 2-noncrossing trees**

*Original research paper. Pages 201–210*

Isaac Owino Okoth

Full paper (PDF, 197 Kb) | Abstract

*i*,

*j*), where

*i*and

*j*are black vertices, in a path from the root. In this paper, we use generating functions to prove a formula that counts 2-noncrossing trees with a black root to take into account the number of white vertices of indegree greater than zero and black vertices. Here, the edges of the 2-noncrossing trees are oriented from a vertex of lower label towards a vertex of higher label. The formula is a refinement of the formula for the number of 2-noncrossing trees that was obtained by Yan and Liu and later on generalized by Pang and Lv. As a consequence of the refinement, we find an equivalent refinement for 2-noncrossing trees with a white root, among other results.

**Book presentation: “Arithmetic Functions”**

*Book presentation. Pages 211–212*

Book presentation (PDF, 324 Kb)

*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-NP2/26/2020.*