Victor Enrique Vizcarra Ruiz

arXiv2025

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

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.

math.GM↗

arXiv2016

Generalization of Numerical Series and its Relationship with the Polynomial Equations and Artithmetic Trapezoids

The close relationship among the polynomial functions and Fibonacci numerical sequences is shown in this paper. These numerical sequences are defined by the recurrence equation $x_{k + n} = \displaystyle\sum_{j = 0}^{n-1}α_j x_{k + j}$, where $n$ is the polynomial degree and $α$'s, the polynomial coefficients. The arithmetic trapezoid resulting from the recurrence equations is also shown. This trapezoid is nothing but a generalization of Pascal's Triangle. Trapezoid is a convenient name because the form it appears does not have the `upper end` of a usual triangle. This study shows that each polynomial generates infinite sequences, and that each sequence generates only a single arithmetic trapezoid.

math.HO↗