A Family of Continued Fraction Identities for Arctangent Values

TL;DR

This paper introduces a new family of continued fraction identities for arctangent values, proving their convergence properties and demonstrating their superiority over traditional series.

Contribution

It establishes a two-parameter family of continued fraction identities for arctangent, linking them to classical Gauss continued fractions and analyzing their convergence rates.

Findings

The identities hold for all positive integer pairs with p ≤ q.

Convergence is geometric with a specific asymptotic rate.

Continued fractions outperform Gregory–Leibniz series in numerical efficiency.

Abstract

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 $-\pi/4$ 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.