A series of Ramanujan, two-term dilogarithm identities and some Lucas series

TL;DR

This paper explores Ramanujan-like series, deriving closed forms involving polylogarithms and inverse hyperbolic functions, establishing new dilogarithm identities, and evaluating Lucas series in closed form.

Contribution

It introduces new two-term dilogarithm identities and evaluates Lucas series, expanding the understanding of special series related to Ramanujan's work.

Findings

Closed form expressions for specific series involving polylogarithms and inverse hyperbolic functions

New identities for two-term dilogarithm functions

Closed-form evaluations of Lucas number series

Abstract

We study an elementary series that can be considered a relative of a series studied by Ramanujan in Part 1 of his Lost Notebooks. We derive a closed form for this series in terms of the inverse hyperbolic arctangent and the polylogarithm. Special cases will follow in terms of the Riemann zeta and the alternating Riemann zeta function. In addition, some trigonometric series will be expressed in terms of the Clausen functions. Finally, a range of new two-term dilogarithm identities will be proved and some difficult series involving Lucas numbers will be evaluated in closed form.