arXiv subjects

Jerico B. Bacani

Publications and source records attributed to Jerico B. Bacani.

3 recordsLinked to original sources

arXiv2015

On Two Nonlinear Difference Equations

The behavior of solutions of the following nonlinear difference equations \[ x_{n+1}=\displaystyle\frac{q}{p+x_n^ν} \quad \text{and} \quad y_{n+1}=\displaystyle\frac{q}{-p+y_n^ν}, \] where $p, q \in\mathbb{R}^+$ and $ν\in \mathbb{N}$ are studied. The solution form of these two equations when $ν=1$ are expressed in terms of Horadam numbers. Furthermore, the behavior of their solutions are investigated for all integer $ν> 0$ and several numerical examples are presented to illustrate the results exhibited. The present work generalizes those seen in [{\it Adv. Differ. Equ.}, {\bf 2013}:174 (2013), 7 pages].

math.DS↗

arXiv2015

On Linear Recursive Sequences with Coefficients in Arithmetic-Geometric Progressions

We present a certain generalization of a recent result of M. I. Cirnu on linear recurrence relations with coefficient in progressions [2]. We provide some interesting examples related to some well-known integer sequences, such as Fibonacci sequence, Pell sequence, Jacobsthal sequence, and the Balancing sequence of numbers. The paper also provides several approaches in solving the linear recurrence relation under consideration. We end the paper by giving out an open problem.

math.NT↗

arXiv2015

On Generalized Fibonacci Numbers

We provide a formula for the $n^{th}$ term of the $k$-generalized Fibonacci-like number sequence using the $k$-generalized Fibonacci number or $k$-nacci number, and by utilizing the newly derived formula, we show that the limit of the ratio of successive terms of the sequence tends to a root of the equation $x + x^{-k} = 2$. We then extend our results to $k$-generalized Horadam ($k$GH) and $k$-generalized Horadam-like ($k$GHL) numbers. In dealing with the limit of the ratio of successive terms of $k$GH and $k$GHL, a lemma due to Z. Wu and H. Zhang [8] shall be employed. Finally, we remark that an analogue result for $k$-periodic $k$-nary Fibonacci sequence can also be derived.

math.NT↗