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 $ \overline{\theta({\tt{t}})} $-Fibonacci polynomial, sequence, and its summation. Here, we have obtained the derivative of the $ \overline{\theta({\tt{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 $ \overline{\theta({\tt{t}})} $-Fibonacci generating function. In addition, we have included appropriate examples to demonstrate our findings by using MATLAB.
Citation: Rajiniganth Pandurangan, Sabri T. M. Thabet, Imed Kedim, Miguel Vivas-Cortez. On the Generalized $ \overline{\theta({\tt{t}})} $-Fibonacci sequences and its bifurcation analysis[J]. AIMS Mathematics, 2025, 10(1): 972-987. doi: 10.3934/math.2025046
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 $ \overline{\theta({\tt{t}})} $-Fibonacci polynomial, sequence, and its summation. Here, we have obtained the derivative of the $ \overline{\theta({\tt{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 $ \overline{\theta({\tt{t}})} $-Fibonacci generating function. In addition, we have included appropriate examples to demonstrate our findings by using MATLAB.
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