Deriving Dilogarithm Identities from the Ratio of Arctangent Integrals

TL;DR

This paper derives new dilogarithm identities and ladders by analyzing ratios of arctangent integrals, extending previous results and proving conjectures in the field.

Contribution

It introduces novel functional dilogarithm equations and ladders, providing analytic proofs and extending known identities using integral ratio methods.

Findings

Derived new 3- and 6-term dilogarithm functional equations.

Proved a conjectured 2-term dilogarithm identity of Bytsko.

Extended results for the Bloch-Wigner function.

Abstract

Building on results by Abouzahra and Lewin, McIntosh, and Kirilov we derive new functional dilogarithm equations and consequent diologarithim ladders. By showing that the ratio of a pair of sextic and cubic integrals equals a rational constant, we construct new 3- and 6-term functional equations, from which we derive an analytic proof of an identity by Loxton-Lewin, as well as a pair of quartic-base dilogarithm ladders, also believed to be new, building on Loxton's result. Finally, we prove conjectured 2-term dilogarithm identities of Bytsko, and extend his result for the Bloch-Wigner function using the above methods.