The Newton approximation, the Hurwitz continued fraction, and the Sierpinski series for relatively quadratic units over certain imaginary quadratic number 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 $\alpha \in \mathbb{C}$ 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 $\alpha$ on the complex plane. We also give an estimate of the error of the Newton approximation of the unit $\alpha$.