Relation Functions Evaluated from Unique Coefficient Patterns

TL;DR

This paper introduces a method to analyze polynomials with specific coefficient patterns, enabling efficient evaluation of coefficients through relation functions derived from unique coefficient patterns.

Contribution

It presents a novel approach to generate and utilize relation functions for polynomials with unique coefficient patterns, simplifying coefficient evaluation.

Findings

Derived relation functions for polynomial coefficient patterns

Enabled direct coefficient computation without full expansion

Applicable to polynomials of the form (x^n + ... + 1)^l

Abstract

In this paper, we study polynomials of the form $f(x)=(x^n+x^{n-1}+...+1)^l$ for $l=1,2,3,4$ to generate a pattern titled "unique coefficient pattern". Namely, we analyze each unique coefficient patterns of $f(x)$ and generate functions titled "relation functions". The approach that we follow will allow us to evaluate desired coefficients for such polynomial expansions by simply using these relation functions.