An Addendum to Plouffe's Ramanujan Identities

TL;DR

This paper introduces a new class of polylogarithm sums with parameters, providing closed-form expressions involving the Dirichlet eta function, and offers an alternative proof for a Ramanujan hyperbolic sum.

Contribution

It extends existing work by defining parameterized polylogarithm sums and connecting them to the Dirichlet eta function, along with a novel proof of a Ramanujan sum.

Findings

New class of polylogarithm sums with two parameters

Closed-form expressions involving Dirichlet eta function

Alternative proof for a Ramanujan hyperbolic sum

Abstract

We introduce a new class of polylogarithm sums closely related to a family studied by L. Vep\v{s}tas in 2010. These generalized sums depend on two free parameters and yield closed-form expressions involving the Dirichlet eta function. Additionally, we present an alternative proof for a hyperbolic sum originally discussed by Ramanujan.