On recurrence relations arising from NRS(2) applied to a cubic polynomial

TL;DR

This paper investigates the properties of recurrence relations from NRS(2) when applied to cubic polynomials, revealing positivity and combinatorial structure in the error terms.

Contribution

It establishes that the leading coefficients of error terms are positive rational functions of the polynomial's zeros and expresses these as sums over new combinatorial objects called radius-value trees.

Findings

Leading coefficients are positive rational functions in zeros.

Error terms can be expressed as sums over radius-value trees.

Provides a combinatorial interpretation of recurrence errors.

Abstract

We prove that the leading coefficient of the "error" terms of NRS(2) applied to a cubic polynomial $f(z)$ with starting point $(-\frac{a_1}{a_2}, -\frac{a_1}{a_2})$ are positive-coefficient rational functions in the zeros of $f(z)$. We express these terms as a sum over combinatorial objects which we call radius-value trees.