A Unified Quadratic Algebraic Approach to Hadamard–Fibonacci Transformations

In this paper, we introduce and study a Hadamard-type transformation associated with quadratic Fibonacci numbers over the parametrized quadratic algebra Kp=R[h]/(h2−p) for p∈R. Rather than generalizing the scalar recurrence relation itself, we change the algebra in which the sequence takes its values and define the quadratic Fibonacci elements by Fn(p):=Fn+hFn+1, inspired by the relation αn=Fnα+Fn−1 for the golden ratio α and the matrix representations of Fibonacci sequences. To proceed, we employ generating functions, spectral operator theory, and the geometry of Kp based on the parameter p. By using Binet-type representations, we obtain a closed formula for the transform of an arbitrary generating function and describe the corresponding operator in a spectral way. In particular, we show that the monomials form an eigenbasis of the Hadamard–Fibonacci operator and that the eigenvalues are exactly the quadratic Fibonacci elements. Furthermore, we examine the preservation of rationality and algebraicity, the scaling of singularities, and the asymptotic growth rate under iteration, accompanied by explicit applications to harmonic numbers, Catalan numbers, and Stirling numbers of the second kind.