A Family of Continued Fraction Identities for Arctangent Values
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917365368750080 |
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| author | Wang, Chao |
| author_facet | Wang, Chao |
| contents | We prove a two-parameter family of continued fraction identities for $\arctan(p/q)$, where $p$ and $q$ are positive integers with $p\le q$. For every such pair, the identity \[
\arctan\frac{p}{q}
= \cfrac{p}{q+\cfrac{p^2}{3q+\cfrac{(2p)^2}{5q+\cfrac{(3p)^2}{7q+\cdots}}}} \] holds, and a sign-flipped variant represents $-\arctan(p/q)$. The proof proceeds by identifying these continued fractions as explicit equivalence transforms of the classical Gauss continued fraction for $\arctan z$. Setting $p=q=1$ recovers a specific identity for $-π/4$ that appeared in the Ramanujan Machine project. We establish that the convergence is geometric with asymptotic rate $(\sqrt{p^2+q^2}-q)^2/p^2$, and we determine the exact threshold at which the Worpitzky criterion applies. Numerical data confirm the theoretical rates and show that the continued fractions dramatically outperform the Gregory--Leibniz series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11892 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Family of Continued Fraction Identities for Arctangent Values Wang, Chao General Mathematics We prove a two-parameter family of continued fraction identities for $\arctan(p/q)$, where $p$ and $q$ are positive integers with $p\le q$. For every such pair, the identity \[ \arctan\frac{p}{q} = \cfrac{p}{q+\cfrac{p^2}{3q+\cfrac{(2p)^2}{5q+\cfrac{(3p)^2}{7q+\cdots}}}} \] holds, and a sign-flipped variant represents $-\arctan(p/q)$. The proof proceeds by identifying these continued fractions as explicit equivalence transforms of the classical Gauss continued fraction for $\arctan z$. Setting $p=q=1$ recovers a specific identity for $-π/4$ that appeared in the Ramanujan Machine project. We establish that the convergence is geometric with asymptotic rate $(\sqrt{p^2+q^2}-q)^2/p^2$, and we determine the exact threshold at which the Worpitzky criterion applies. Numerical data confirm the theoretical rates and show that the continued fractions dramatically outperform the Gregory--Leibniz series. |
| title | A Family of Continued Fraction Identities for Arctangent Values |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2601.11892 |