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

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 $ \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.

Keywords

• generalized nabla operator variable coefficients
• fibonacci sequence
• fibonacci summation
• proportional $ \alpha $-derivative
• bifurcation