Volume 30, 2024, Number 2 (Online First)

Volume 30 ▶ Number 1 ▷ Number 2 (Online First)

Volume opened: 4 May 2024
Status: In progress

MIN-turns and MAX-turns in k-Dyck paths: A pure generating function approach
Original research paper. Pages 213–222
Helmut Prodinger
Full paper (PDF, 188 Kb) | Abstract

$k$ -Dyck paths differ from ordinary Dyck paths by using an up-step of length $k$ . We analyze at which level the path is after the $s$ -th up-step and before the $(s+1)$ -st up-step. In honour of Rainer Kemp who studied a related concept 40 years ago, the terms max-terms and min-terms are used. Results are obtained by an appropriate use of trivariate generating functions; practically no combinatorial arguments are used.

Some new results on the negative polynomial Pell’s equation
Original research paper. Pages 223–235
K. Anitha, I. Mumtaj Fathima, and A. R. Vijayalakshmi
Full paper (PDF, 259 Kb) | Abstract

We consider the negative polynomial Pell’s equation $P^2(X)-D(X)Q^2(X)=-1$ , where $D(X)\in \mathbb{Z}[X]$ be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions $P(X), \, Q(X)$ with integer coefficients.

Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, II
Original research paper. Pages 236–252
A. M. S. Ramasamy
Full paper (PDF, 284 Kb) | Abstract

Let $\rho$ be an odd prime $\ge 11.$ In Part I, starting from an $M$ -cycle in a finite field $\mathbb{F}_\rho$ , we have established how the divisors of Mersenne, Fermat and Lehmer numbers arise. The converse question is taken up in this Part with the introduction of an arithmetic function and the notion of a split-associated prime.

On some identities for the DGC Leonardo sequence
Original research paper. Pages 253–270
Çiğdem Zeynep Yılmaz and Gülsüm Yeliz Saçlı
Full paper (PDF, 261 Kb) | Abstract

In this study, we examine the Leonardo sequence with dual-generalized complex ( $\mathcal{DGC}$ ) coefficients for $\mathfrak{p} \in \mathbb R$ . Firstly, we express some summation formulas related to the $\mathcal{DGC}$ Fibonacci, $\mathcal{DGC}$ Lucas, and $\mathcal{DGC}$ Leonardo sequences. Secondly, we present some order- $2$ characteristic relations, involving d’Ocagne’s, Catalan’s, Cassini’s, and Tagiuri’s identities. The essential point of the paper is that one can reduce the calculations of the $\mathcal{DGC}$ Leonardo sequence by considering $\mathfrak{p}$ . This generalization gives the dual-complex Leonardo sequence for $\mathfrak p =-1$ , hyper-dual Leonardo sequence for $\mathfrak p =0$ , and dual-hyperbolic Leonardo sequence for $\mathfrak p =1$ .

The generalized order (k,t)-Mersenne sequences in groups
Original research paper. Pages 271–282
E. Mehraban, Ö. Deveci, E. Hincal
Full paper (PDF, 302 Kb) | Abstract

The purpose of this paper is to determine the algebraic properties of finite groups via a Mersenne-like sequence. Firstly, we introduce the generalized order $(k,t)$ -Mersenne number sequences and study the periods of these sequences modulo $m$ . Then, we get some interesting structural results. Furthermore, we expand the generalized order $(k,t)$ -Mersenne number sequences to groups and we give the definition of the generalized order $(k,t)$ -Mersenne sequences, $MQ_k^t(G,X)$ , in the $j$ -generator groups and also, investigate these sequences in the non-Abelian finite groups in detail. At last, we obtain the periods of the generalized order $(k,t)$ -Mersenne sequences in some special groups as applications of the results produced.

Some infinite series summations involving linear recurrence relations of order 2 and 3
Original research paper. Pages 283–310
Anthony G. Shannon, Peter J.-S. Shiue, Shen C. Huang, Ali Balooch, Yu-Chung Liu
Full paper (PDF, 319 Kb) | Abstract

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$ , including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$ . The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

The error term of the sum of digital sum functions in arbitrary bases
Original research paper. Pages 311–318
Erdenebileg Erdenebat and Ka Lun Wong
Full paper (PDF, 219 Kb) | Abstract

Let $k$ be a non-negative integer and $q > 1$ be a positive integer. Let $s_q(k)$ be the sum of digits of $k$ written in base $q.$ In 1940, Bush proved that $A_q(x)=\sum_{k \leq x} s_q (k)$ is asymptotic to $\frac{q-1}{2}x \log_q x.$ In 1968, Trollope proved an explicit formula for the error term of $A_2(n-1),$ labeled by $-E_2(n),$ where $n$ is a positive integer. In 1975, Delange extended Trollope’s result to an arbitrary base $q$ by another method and labeled the error term $nF_q(\log_q n).$ When $q=2,$ the two formulas of the error term are supposed to be equal, but they look quite different. We proved directly that those two formulas are equal. More interestingly, Cooper and Kennedy in 1999 applied Trollope’s method to extend $-E_2(n)$ to $-E_q(n)$ with a general base $q,$ and we also proved directly that $nF_q(\log_q n)$ and $-E_q(n)$ are equal for any $q.$

On some series involving the binomial coefficients $\binom{3n}{n}$
Original research paper. Pages 319–334
Kunle Adegoke, Robert Frontczak and Taras Goy
Full paper (PDF, 280 Kb) | Abstract

Using a simple transformation, we obtain much simpler forms for some series involving binomial coefficients $\binom{3n}n$ derived by Necdet Batır. New evaluations are given and connections with Fibonacci numbers and the golden ratio are established. Finally we derive some Fibonacci and Lucas series involving the reciprocals of $\binom{3n}n$ .

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