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

TL;DR

This paper explores the connections between polynomial functions, Fibonacci sequences, and generalized arithmetic trapezoids, revealing how polynomial degree influences sequence generation and the structure of these trapezoids.

Contribution

It introduces a generalized arithmetic trapezoid derived from polynomial recurrence relations, linking polynomial degree to sequence and trapezoid structures.

Findings

Polynomial sequences are generated by recurrence equations.

Each polynomial produces a unique infinite sequence.

Sequences correspond to a specific generalized trapezoid.

Abstract

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 $x_{k + n} = \displaystyle\sum_{j = 0}^{n-1}\alpha_j x_{k + j}$, where $n$ is the polynomial degree and $\alpha$'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.