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.

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