Study of particular polynomials defined by recurrent linear sequences with applications

Abstract

The primary goal of this research is to delve into the wel l-known tribonacci sequence, explor- ing its inherent identities. The aim is to extend and generalize these findings to encompass a broader class of sequences and their associated polynomials. Within this investigation, we develop a clear and explicit formula for the generalized tribonacci polynomials. Furthermore, we systematical ly establish and examine various properties inherent to these polynomials. Through this exploration, the research seeks to contribute to a deeper understanding of the generalized tribonacci polynomials and their properties. Similarly, we investigate Jacobsthal polynomials (Jn (ρ))n∈N, We also describe some of the properties of Jacobsthal polynomials. after that we are present an alternative formula for both generalized tribonacci polynomi- als (Pn (ρ, ϱ, ϖ))n∈N and generalized tribonacci-Lucas polynomials (Ln(x, y, z))n∈N. These formulas are derived through the application of combinatorial calculus, along with an ex-

amination of their summation. Additional ly, we provide explicit expressions for the partial derivatives of these polynomials, denoted as ∂Pn (ρ, ϱ, ϖ) and ∂Ln (ρ, ϱ, ϖ), concerning one of their variables. Furthermore, we discuss various properties associated with these polyno- mials and their derivatives. Final ly we introduce the generalized tribonacci and generalized tribonacci Quaternions like polynomial. We commence by establishing several fundamental identities related to these quaternions. Subsequently, we derive Binet’s formula, generating functions, and a summa- tion formula specifical ly tailored for this class of quaternions.