Functional Dilogarithm Identities in Quadratic Fields

Abstract

We derive three- and six-term functional dilogarithm identities whose arguments lie in $\mathbb{Q}(u,\sqrt{4-3u^2})$ and $\mathbb{Q}(u,\sqrt{u(4-3u)})$. Our approach is based on an integral-to-${}_4F_3$ correspondence that converts families of cubic and sextic integrals into hypergeometric identities, providing a systematic method for constructing functional equations for the dilogarithm over quadratic fields. We demonstrate the power of this method by giving an analytic proof of the classical Loxton--Lewin $2\cos(4\pi/9)$ identity, deriving a new family of dilogarithm ladders of quartic base lying in $\mathbb{Q}(\sqrt{33})$, and proving conjectural two-term identities of Bytsko. As a further application, we obtain rapidly convergent ${}_4F_3$ series for $\mathrm{Cl}_2(\pi/3)$ and explicit relations connecting $\mathbb{Q}(\sqrt{13})$ and $\mathbb{Q}(\sqrt{3})$. Finally, a PSLQ-based search over palindromic quartic units yields new ladder relations with arguments built from $2\tan(\pi/8)\cos(\pi/5)$ and $\tan(3\pi/20)$, analogous to known trigonometric identities of Watson, Loxton, and Gordon--McIntosh.