A Dilogarithm Series and a Generalization of a Theorem of Bridgeman

TL;DR

This paper derives a new two-parameter series identity for the Rogers dilogarithm using Abel's relation, generalizing Bridgeman's identities related to Pell's equations through a novel approach.

Contribution

It introduces a new series identity for the Rogers dilogarithm and generalizes Bridgeman's results using Abel's five-term relation and Lucas sequences.

Findings

Derived a new two-parameter series identity for the Rogers dilogarithm.

Obtained dilogarithm series involving Lucas sequences.

Generalized Bridgeman's series identities related to Pell's equations.

Abstract

Using Abel's five-term relation, we derive a new two-parameter series identity for the Rogers dilogarithm. By specializing this identity, we obtain dilogarithm series involving Lucas sequences. These results generalize certain series identities of Bridgeman related to solutions of Pell's equations, which were obtained via a completely different approach.