On the leading and penultimate leading coefficients for NRS(2) applied to a cubic polynomial

Abstract

We prove that the leading and penultimate leading coefficients in $u_3$ of the ``error" terms of NRS(2) applied to a cubic polynomial $f(z)={\sum }_{i=0}^3{a}_i{z}^i={\prod }_{i=1}^3(1-u_{i}z)$ with starting point $(-\frac{a_1}{a_2},-\frac{a_1}{a_2})$ are positive-coefficient polynomials in $u_1$ and $u_2$. Our proof for the leading coefficients simplifies that of \cite{DeFranco} and extends to the penultimate leading coefficients as well.