On the Generalized $\overline{\theta (\mathtt{t})}$-Fibonacci sequences and its bifurcation analysis

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 $\overline{\theta (\mathtt{t})}$-Fibonacci polynomial, sequence, and its summation. Here, we have obtained the derivative of the $\overline{\theta (\mathtt{t})}$-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 $\overline{\theta (\mathtt{t})}$-Fibonacci generating function. In addition, we have included appropriate examples to demonstrate our findings by using MATLAB.