2026/04/27 by Cetin Hakimoglu-Brown
#math.CA
We derive three- and six-term functional dilogarithm identities whose arguments lie in ℚ(u,√(4-3u2)) and ℚ(u,√(u(4-3u))). Our approach is based on an integral-to-4F3 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 2cos(4π/9) identity, deriving a new family of dilogarithm ladders of quartic base lying in ℚ(√(33)), and proving conjectural two-term identities of Bytsko. As a further application, we obtain rapidly convergent 4F3 series for Cl2(π/3) and explicit relations connecting ℚ(√(13)) and ℚ(√(3)). Finally, a PSLQ-based search over palindromic quartic units yields new ladder relations with arguments built from 2tan(π/8)cos(π/5) and tan(3π/20), analogous to known trigonometric identities of Watson, Loxton, and Gordon--McIntosh.