Gaussian Balancing-Like Sequences and Their Asymptotic Geometry

In this study, we investigate complex extensions of balancing-like sequences. These sequences are defined by second-order linear recurrence relations with complex initial conditions and depend on a real parameter A > 2. We first derive explicit Binet-type formulas, generating functions, and matrix representations for sequences. Several fundamental identities, including Cassini, Catalan, d'Ocagne-type, and mixed identities, are also established.

A central aim of this study is to examine not only the algebraic structure of these sequences but also their geometric behavior in the complex plane. Using the dominant root of the characteristic equation

λ² − Aλ + 1 = 0,

we show that the terms of these sequences become asymptotically aligned along the same ray in the complex plane. More precisely, their arguments converge to a common limiting direction determined by the dominant characteristic root. Furthermore, the angular deviation from this limiting direction decreases exponentially.

These results show that the proposed extensions preserve many classical algebraic properties of balancing-type sequences while also exhibiting a well-defined asymptotic geometry. Thus, the study provides a bridge between recurrence sequence theory, complex valued sequences, and discrete dynamical behavior.

The methods of the study rely on characteristic-root analysis, Binet-type formulas, generating functions, and companion matrix representations. These tools allow us to obtain closed forms, prove algebraic identities, and analyze the asymptotic direction of the sequences in the complex plane.