Analytic Dilogarithm Identities

Abstract

We introduce dilogarithm identities through a beta integral-based technique that we apply to provide analytic proofs of previously conjectured dilogarithm relations, solving open problems given by both Bytsko and Campbell, and that we further apply to construct and prove new ladder relations with quartic and sextic bases. We also apply our method to introduce and prove two-term $\operatorname{Li}_2$-relations and ladder-like identities with arguments in algebraic number fields such as $\mathbb{Q}(\sqrt{2})$, $\mathbb{Q}(\sqrt{3})$, $\mathbb{Q}(\sqrt{5})$, and $\mathbb{Q}(\sqrt{-7})$. Moreover, single-term dilogarithm evaluations are introduced and derived.