On h(x)-Fibonacci polynomials in an arbitrary algebra

TL;DR

This paper introduces h(x)-Fibonacci polynomials within arbitrary finite-dimensional algebras over real or complex fields, generalizing existing quaternion and octonion polynomial forms, and establishes various fundamental identities and formulas.

Contribution

It generalizes h(x)-Fibonacci polynomials to arbitrary algebras and derives key identities and formulas for these generalized polynomials.

Findings

Derived summation formula for h(x)-Fibonacci polynomials

Established generating function and Binet-style formula

Proved Catalan and d'Ocagne-type identities

Abstract

In this paper, we introduce h(x)-Fibonacci polynomials in an arbitrary finite-dimensional unitary algebra over a field K (K = R,C), which generalize both h(x)-Fibonacci quaternion polynomials and h(x)-Fibonacci octonion polynomials. For h(x)-Fibonacci polynomials in such an arbitrary algebra, we prove summation formula, generating function, Binet-style formula, Catalan-style identity, and d Ocagne-type identity.