Some identities of generalized Tribonacci and Jacobsthal polynomials

Abstract

In this study, we denote $(t'_{n}(x))_{n\in \mathbb{N}}$ the generalized Tribonacci polynomials, which are defined by $t'_{n}(x)=x^{2}t'_{n-1}(x)+xt'_{n-2}(x)+t'_{n-3}(x), n \geqslant 4,$ with $t_{1}(x)=a, t_{2}(x)=b, t_{3}(x)=cx^{2}$ and we drive an explicit formula of $(t'_{n}(x))_{n\in \mathbb{N}}$ in terms of their coefficients $T'(n,j)$, Also, we establish some properties of $(t_{n}(x))_{n\in \mathbb{N}}$. Similarly, we study the Jacobsthal polynomials $(J_{n}(x))_{n\in \mathbb{N}}$, where $J_{n}(x)=J_{n-1}(x)+x J_{n-2}(x)+ x^{2} J_{n-3}(x), n \geqslant 4$, with $J_{1}(x)= J_{2}(x)=1, J_{3}(x)=x+1$ and describe some properties.