On refinements of two-term Machin-like formulas

TL;DR

This paper introduces a refinement process for two-term Machin-like formulas using continued fractions, generating sequences of formulas with rational arguments that converge to pi/4 with geometric decay.

Contribution

It develops a novel method leveraging continued fractions to generate and analyze sequences of two-term Machin-like formulas with rational arguments.

Findings

Sequences of formulas converge to pi/4 with geometric decay.

Explicit formulas and estimates for coefficients and arguments.

Method demonstrated with Euler's Machin-like formula.

Abstract

We develop a refinement process for two-term Machin-like formulas: $a_0 \arctan{u_0} + a_1 \arctan{u_1} = \frac{\pi}{4}$ (where $a_0 , a_1 \in \mathbb{Z}$, $u_0 , u_1 \in \mathbb{Q}_+^*$, $u_0 > u_1$) by exploiting the continued fraction expansion of the ratio $\alpha := \frac{\arctan{u_0}}{\arctan{u_1}}$. This construction yields a sequence of derived two-term Machin-like formulas: $a_{- n} \arctan{u_n} + a_{- n + 1} \arctan{u_{n + 1}} = \frac{\pi}{4}$ ($n \in \mathbb{N}$) with positive rational arguments $u_n$ decreasing to zero and corresponding integer coefficients $a_{- n}$. We derive closed forms and estimates for $a_{-n}$ and $u_n$ in terms of the convergents of $\alpha$ and prove that the associated rational sequence $(a_{- n} u_n + a_{- n + 1} u_{n + 1})_n$ converges to $\pi/4$ with geometric decay. The method is illustrated using Euler's two-term Machin-like formula : $\arctan(1/2) + \arctan(1/3) = \pi/4$.