Explicit Identities for Horadam Polynomials: Generalized Fibonacci Formulations and Special Cases
Abstract
In this paper, we investigate the generalized Fibonacci (Horadam) polynomials and concentrate on two special subclasses, which we introduce as the (r,s)-Fibonacci and (r,s)-Lucas polynomials. Our primary aim is to present and establish several identities that connect these two families, thereby extending classical relations between Fibonacci and Lucas sequences into a broader polynomial framework. The identities obtained not only highlight the structural interplay between the (r,s)-Fibonacci and (r,s)-Lucas polynomials but also enrich the theory of generalized Horadam polynomials by revealing new algebraic connections. This work is devoted exclusively to the derivation and exposition of such identities, providing a foundation for further exploration of recurrence-based polynomial structures.
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References
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