Some Properties of the Product Sequences Involving Fibonacci Numbers

A. Let Fn denote the n-th Fibonacci number for n N.For every m N, we define the P -Fibonacci sequence Pm,n by the recurrenceIn this paper, we investigate the structure and properties of the P-Fibonacci sequences, which arise from specific algebraic combinations of Fibonacci numbers.Through analytical exploration and pattern recognition, we uncover and prove several intriguing identities related to these sequences.We also present illustrative examples to highlight their recurring behaviors.This study not only contributes to the theoretical understanding of Fibonacci-related sequences but also lays groundwork for potential applications and further research in combinatorics, number theory, and algorithmic design.