Partial Fraction Expansions for Newton's and Halley's Iterations for Square Roots

TL;DR

This paper derives a formula for rational functions generated by Newton's and Halley's methods when approximating square roots, linking them to Chebyshev's polynomials, and confirms a conjecture about their coefficients.

Contribution

It provides a new explicit formula for these rational functions in the square root case, connecting them to Chebyshev's polynomials and proving a conjecture on coefficient signs.

Findings

Derived a formula for rational functions in the square root case

Linked rational functions to Chebyshev's polynomials

Confirmed the sign of coefficients as conjectured

Abstract

When Newton's method, or Halley's method is used to approximate the $p${th} root of $1-z$, a sequence of rational functions is obtained. In this paper, a beautiful formula for these rational functions is proved in the square root case, using an interesting link to Chebyshev's polynomials. It allows the determination of the sign of the coefficients of the power series expansion of these rational functions. This answers positively the square root case of a proposed conjecture by Guo(2010).