A New Three-Parameter Family of Non-Symmetric Fibonacci Polynomials and Associated Structural Formulas

This paper introduces and investigates a family of non-symmetric Fibonacci polynomials (NSFPs) characterized by three parameters, generalizing well-known variants. We develop a series of structural formulas for these polynomials that elucidate their relationship with classical and certain generalized Fibonacci-type sequences. Additionally, we derive the generating function, a Binet-type representation, a connection formula with classical Fibonacci polynomials, and an explicit series expression. Furthermore, we establish standard and mixed linearization formulas, derivative expansions as linear combinations of classical Fibonacci and Lucas polynomial bases, and moment formulas for derivatives. We also deduce derivative expansions in terms of several symmetric and non-symmetric polynomial bases, using the corresponding classical inversion formulas. We extend the NSFPs to negative indices, yielding corresponding connection and linearization identities. In addition, the negative-to-positive connection is used to eliminate negative-index terms from the first-derivative expansion and to construct a finite first-order differentiation operational matrix for the NSFP basis. The resulting identities broaden the structural framework of the NSFPs and provide explicit tools for their future use in approximation methods and numerical algorithms.