On refinements of two-term Machin-like formulas

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 $a_0,a_1\in \mathbb{Z}$, $u_0,u_1\in {\mathbb{Q}}_{+}^{\ast }$, $u_0>u_1$) 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 $\pi \mathrm{/}4$ with geometric decay. The method is illustrated using Euler’s two-term Machin-like formula : $\arctan (1\mathrm{/}2)+\arctan (1\mathrm{/}3)=\pi \mathrm{/}4$.