The Jacobsthal–Padovan–Fibonacci p-sequence and its application in the concise representation of vibrating systems with dual proximal groups

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发表日期：

2024年11月22日

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Özgür Erdağ,
James F. Peters,
Ömür Deveci

摘要

The focus of this paper is on the Jacobsthal–Padovan–Fibonacci (JPF) p-sequence $\left\{ JPa_{n}^{F,p}\right\}$ $\{J, P, a_{n}^{F, p}\}$ , $p\ge 3$ $p \geq 3$ , $n>0$ $n > 0$ and the connections between the Fibonacci and Jacobsthal–Padovan p-sequences. Also, a new Binet formula and a new combinatorial representation as well as some useful properties of the JPF p-numbers are given. The main results are an exponential form of $JPa_{n}^{F,p}$ $J P a_{n}^{F, p}$ , useful identities for $m\ge p+4$ $m \geq p + 4$ , $P_{m,p}^{JPa,F}$ $P_{m, p}^{J P a, F}$ , $R_{m,p}^{JPa,F}$ $R_{m, p}^{J P a, F}$ and for $m>p+4$ $m > p + 4$ , $S_{m,p}^{JPa,F}$ $S_{m, p}^{J P a, F}$ , topologies for every descriptive proximity space, with application in dual proximal group representations of the topology on a proximity space on the Hilbert envelope lobes on a vibrating system waveform.