On the Generalized θ(t)-Fibonacci sequences and its bifurcation analysis

Rajiniganth Pandurangan; Sabri T.M. Thabet; Imed Kedim; Miguel Vivas-Cortez

doi:10.3934/MATH.2025046

On the Generalized θ(t)-Fibonacci sequences and its bifurcation analysis

Rajiniganth Pandurangan^*
, Sabri T.M. Thabet^*
, Imed Kedim
, Miguel Vivas-Cortez^*

^*Corresponding author for this work

Faculty of Exact, Natural, and Environmental Sciences

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	972-987
Number of pages	16
Journal	AIMS Mathematics
Volume	10
Issue number	1
DOIs	https://doi.org/10.3934/MATH.2025046
State	Published - 2025

Bibliographical note

Publisher Copyright:
© 2025 the Author(s), licensee AIMS Press.

Funding

“La derivada fraccional generalizada, nuevos resultados y aplicaciones a desigualdades integrales” Cod UIO-077-2024. This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).

Funders	Funder number
Prince Sattam Bin Abdulaziz University	PSAU/2025/R/1446

Keywords

bifurcation
Fibonacci sequence
Fibonacci summation
generalized nabla operator variable coefficients
proportional α-derivative

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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.",

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AB - 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.

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KW - Fibonacci summation

KW - generalized nabla operator variable coefficients

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On the Generalized θ(t)-Fibonacci sequences and its bifurcation analysis

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