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The Newton approximation, the Hurwitz continued fraction, and the Sierpinski series for relatively quadratic units over certain imaginary quadratic number fields

Asaki Saito, Jun-Ichi Tamura

TL;DR

The paper extends real-analytic identities to complex settings by linking the Newton iteration on $f(X)=X^2-TX+U$ with the Sierpinski series and Hurwitz continued fractions for relatively quadratic units over imaginary quadratic fields, using a two-variable rational-function framework in $T$ and $U$. It proves explicit equalities among the Newton iterates $F^{(n+1)}(0)$, the truncated Sierpinski series $S_n(T,U)$, and truncated Hurwitz CFs, and similarly for $F^{(n+1)}(T)$, with sharp convergence bounds and Eisenstein-field analogues. The results unify three representations of algebraic complex numbers and provide explicit error controls, extending rapidly convergent representations to both Gaussian and Eisenstein settings. This creates a computational bridge among Newton methods, complex continued fractions, and Sierpinski-type series for relatively quadratic units in these fields.

Abstract

The objective of this paper is to show (a)=(b)=(c) as rational functions of $T$, $U$ for (a), (b), (c) given by (a) continued fractions of length $2^{n+1}-1$ with explicit partial denominators in $\left\{-T,U^{-1}T\right\}$, (b) truncated series $\sum_{0\le m\le n} \left(U^{2^m}/\left(h_0(T)h_1(T,U) \cdots h_m(T,U)\right)\right)$ with $h_n$ defined by $h_0:=T$ and $h_{n+1}(T,U):=h_n(T,U)^2-2U^{2^n} (n \geq 0)$, (c) $(n+1)$-fold iteration $F^{(n+1)}(0)= F^{(n+1)}(0,T,U)$ of $F(X)= F(X,T,U) :=X-f(X)/\frac{df}{dX}(X)$ for $f(X)=X^2-T X+U$, and to find explicit equalities among truncated Hurwitz continued fraction expansion of relatively quadratic units $α\in \mathbb{C}$ over imaginary quadratic fields $\mathbb{Q}\left(\sqrt{-1}\right)$, $\mathbb{Q}\left(\sqrt{-3}\right)$, rapidly convergent complex series called the Sierpinski series, and the Newton approximation of $α$ on the complex plane. We also give an estimate of the error of the Newton approximation of the unit $α$.

The Newton approximation, the Hurwitz continued fraction, and the Sierpinski series for relatively quadratic units over certain imaginary quadratic number fields

TL;DR

The paper extends real-analytic identities to complex settings by linking the Newton iteration on with the Sierpinski series and Hurwitz continued fractions for relatively quadratic units over imaginary quadratic fields, using a two-variable rational-function framework in and . It proves explicit equalities among the Newton iterates , the truncated Sierpinski series , and truncated Hurwitz CFs, and similarly for , with sharp convergence bounds and Eisenstein-field analogues. The results unify three representations of algebraic complex numbers and provide explicit error controls, extending rapidly convergent representations to both Gaussian and Eisenstein settings. This creates a computational bridge among Newton methods, complex continued fractions, and Sierpinski-type series for relatively quadratic units in these fields.

Abstract

The objective of this paper is to show (a)=(b)=(c) as rational functions of , for (a), (b), (c) given by (a) continued fractions of length with explicit partial denominators in , (b) truncated series with defined by and , (c) -fold iteration of for , and to find explicit equalities among truncated Hurwitz continued fraction expansion of relatively quadratic units over imaginary quadratic fields , , rapidly convergent complex series called the Sierpinski series, and the Newton approximation of on the complex plane. We also give an estimate of the error of the Newton approximation of the unit .
Paper Structure (7 sections, 12 theorems, 85 equations, 4 figures)

This paper contains 7 sections, 12 theorems, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hurwitz Continued fraction expansion
  3. The Newton iterator and the Sierpinski series
  4. The Newton iterator and Continued fractions
  5. The speed of Convergence
  6. Main results
  7. Relatively quadratic units over $K_E$

Key Result

Lemma 2.1

Let $G_1(u)$ ($u \in \left\{\pm 1,\pm \sqrt{-1}\right\}$) be a subset of $R_G$ defined by Let $f(X)=f(X;t,u):=X^2-t X+u\in R_G[X]$ be a polynomial with $t, u$ satisfying Then $f$ has distinct roots $\alpha$, $\beta$ with $|\alpha|<1<|\beta|$.

Figures (4)

  • Figure 1: (a) $G_2(1)$, (b) $G_2\left(\sqrt{-1}\right)$, (c) $G_2(-1)$, and (d) $G_2\left(-\sqrt{-1}\right)$.
  • Figure 2: (a) $G_3(1)=G_3(-1)$, (b) $G_3\left(\sqrt{-1}\right)$, and (c) $G_3\left(-\sqrt{-1}\right)$.
  • Figure 3: (a) $E_2\left(b^0\right)$, (b) $E_2\left(b^1\right)$, (c) $E_2\left(b^2\right)$, (d) $E_2\left(b^3\right)$, (e) $E_2\left(b^4\right)$, and (f) $E_2\left(b^5\right)$.
  • Figure 4: $E_{*}$

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Remark 2.4
  • proof : Proof of Lemma \ref{['Lem:OpenDiscsIncludingRoots']}
  • Lemma 2.5
  • Remark 2.6
  • proof : Proof of Lemma \ref{['Lem:HCFGExpansions']}
  • Lemma 3.1
  • ...and 13 more