@article{Pandurangan2025, 
author = {Rajiniganth Pandurangan and Sabri T. M. Thabet and Imed Kedim and Miguel Vivas-Cortez},
title = {On the Generalized              θ      (                        t                    )        ¯  -Fibonacci sequences and its bifurcation analysis},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {1},
pages = {972-987},
keywords = {bifurcation, Fibonacci sequence, generalized nabla operator variable coefficients, Fibonacci summation, proportional α-derivative},
url = {https://www.sciopen.com/article/10.3934/math.2025046},
doi = {10.3934/math.2025046},
abstract = {This paper introduces a general nabla operator of order two that includes coefficients of various trigonometric functions. We also introduce its inverse, which leads us to derive the second-order              θ      (                        t                    )        ¯  -Fibonacci polynomial, sequence, and its summation. Here, we have obtained the derivative of the              θ      (                        t                    )        ¯  -Fibonacci polynomial using a proportional derivative. Furthermore, this study presents derived theorems and intriguing findings on the summation of terms in the second-order Fibonacci sequence, and we have investigated the bifurcation analysis of the              θ      (                        t                    )        ¯  -Fibonacci generating function. In addition, we have included appropriate examples to demonstrate our findings by using MATLAB.}
}