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

2016/07/20 by Victor Enrique Vizcarra Ruiz, Ruiz, Victor Enrique Vizcarra
Physics and Astronomy · Mathematics · #Advanced Mathematical Theories and Applications #Mathematics and Applications #Experimental and Theoretical Physics Studies

paper · pdf · doi:10.48550/arxiv.1607.06002

Abstract

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

Related

• Algebraic relations for recursive sequences
• On certain Fibonacci representations
• On Higher Dimensional Fibonacci Numbers, Chebyshev Polynomials and\n Sequences of Vector Convergents
• Infinite Series Involving Fibonacci Numbers Via Apéry-Like Formulae
• Automating John P. D'Angelo's method to study Complete Polynomial Sequences