Generalization of Binet's formula for Fibonacci-type numeric sequences through the use of arithmetic pseudo-operators

Abstract

This paper presents an innovative approach to the study of recurrent sequences by introducing the concept of arithmetic pseudo-operators. Unlike conventional operators, these pseudo-operators are pure complex numbers with specific structural properties, allowing for unprecedented operational reformulations. Represented by the symbols ``$+$'', ``$/$'' (slash), ``$\setminus$'' (aslash), ``$\bot$'', ``$\top$'', and ``$\dashv$'', these operators correspond to rotations in the unit circle of the complex plane and generalize fundamental operations, as seen in the identities ``$1 / 1 \setminus 1 = 0$'' and ``$1 \bot 1 \top 1 \dashv 1 = 0$'', which exhibit behavior analogous to conventional subtraction ``$1 - 1 = 0$''. Based on this structure, we reformulate Binet's equations for the Fibonacci and Tribonacci sequences and outline the path for their generalization to the Tetranacci sequence \cite{koshy}. This new perspective not only enhances the understanding of higher-order recurrences but also suggests potential applications in discrete mathematics and computational algebra, expanding the scope of classical algebraic operations.