The Newton approximation, the Hurwitz continued fraction, and the Sierpinski series for relatively quadratic units over certain imaginary quadratic number fields
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| Format: | Preprint |
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2025
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| _version_ | 1866911216558931968 |
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| author | Saito, Asaki Tamura, Jun-Ichi |
| author_facet | Saito, Asaki Tamura, Jun-Ichi |
| contents | 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 $α$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_15498 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Newton approximation, the Hurwitz continued fraction, and the Sierpinski series for relatively quadratic units over certain imaginary quadratic number fields Saito, Asaki Tamura, Jun-Ichi Number Theory Primary 11A55, 30B70, Secondary 49M15 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 $α$. |
| title | The Newton approximation, the Hurwitz continued fraction, and the Sierpinski series for relatively quadratic units over certain imaginary quadratic number fields |
| topic | Number Theory Primary 11A55, 30B70, Secondary 49M15 |
| url | https://arxiv.org/abs/2510.15498 |