φ ( x ) -Tribonnaci polynomial‎, ‎numbers‎, ‎and its sum

‎This study presents a general third-order nabla difference operator that allows us to get \(\overline{\varphi({\tt{x}})}\)-Tribonacci sequences‎, ‎Tribonacci numbers‎, ‎and their sum using the coefficients of different trigonometric functions and their‎ ‎inverse‎. ‎In this section‎, ‎we examined the numerical solutions and \(C^*\)-solutions of the‎ ‎\(\overline{\varphi({\tt{x}})}\)-Tribonacci sequences for different functions‎. ‎In addition‎, ‎some interesting conclusions and theorems are obtained for the sum of the terms of the Tribonacci‎ ‎sequence‎. ‎Also‎, ‎we offer appropriate examples to show how to use MATLAB to demonstrate our results‎. ‎