On a series of Ramanujan, dilogarithm values, and solitons

TL;DR

This paper explores advanced properties of dilogarithms, including their extension to complex domains, connections to Bernoulli polynomials, and applications to solitons, with improvements to existing mathematical representations.

Contribution

It introduces new insights into dilogarithm functions, extending their applicability and relating them to other mathematical structures like Bernoulli polynomials and solitons.

Findings

Extended dilogarithm to complex numbers beyond the unit disk

Connected polylogarithms to Bernoulli polynomials

Improved existing representations of polylogarithms

Abstract

We discuss several topics related to polylogarithms with focus on dilogarithms. The topics are: a generating function with harmonic numbers coming from Ramanujan, extending the dilogarithm to complex numbers beyond the unit disk, and relating polylogarithms to Bernoulli polynomials and to solitons. Several improvements and corrections are made to existing representations.