Analytic Dilogarithm Identities

TL;DR

This paper develops a beta integral-based method to prove and discover dilogarithm identities, solving open conjectures and extending identities to algebraic number fields with new evaluations.

Contribution

It introduces a novel beta integral technique for proving and deriving dilogarithm identities, including solutions to open problems and new identities over algebraic number fields.

Findings

Proved conjectured dilogarithm relations using the new method.

Constructed new ladder relations with quartic and sextic bases.

Derived single-term dilogarithm evaluations.

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.