A Family of Continued Fraction Identities for Arctangent Values

We prove a two-parameter family of continued fraction identities for $\arctan (p\mathrm{/}q)$, where $p$ and $q$ are positive integers with $p\le q$. For every such pair, the identity $$\arctan \frac{p}{q}={\displaystyle \frac{p}{q+{\displaystyle \frac{p^2}{3q+{\displaystyle \frac{(2p{)}^2}{5q+{\displaystyle \frac{(3p{)}^2}{7q+\cdots }}}}}}}}$$ holds, and a sign-flipped variant represents $-\arctan (p\mathrm{/}q)$. 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 $-\pi \mathrm{/}4$ that appeared in the Ramanujan Machine project. We establish that the convergence is geometric with asymptotic rate $(\sqrt{p^2+q^2}-q{)}^2\mathrm{/}{p}^2$, 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.