Closed-form solutions of a nonlinear bidimensional difference system via generalized Fibonacci sequences

This paper presents a new model for a two-dimensional nonlinear difference system that incorporates symmetric interactions between two sequences through a scaling parameter $d$ and a continuous one-to-one transformation function $f$. Explicit analytical solutions are derived, establishing a direct connection with the $d$-Fibonacci sequence. The transformation function $f$ plays a crucial role: It accommodates diverse nonlinear iteration patterns and provides a natural mechanism for regulating both growth dynamics and sequence interactions. Moreover, the use of a continuous one-to-one function guarantees that the analytical solutions of transformed systems can be recovered through its inverse mapping. The approach highlights a unified framework linking generalized Fibonacci-type recursions with nonlinear transformations, offering new insights into the structure and solvability of higher-order discrete systems. Several illustrative examples are provided to support the theoretical findings.