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

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.