Closed-form evaluations of log-sine integrals and Ap\'ery-like sums in terms of polylogarithms

TL;DR

This paper introduces a systematic method to evaluate log-sine integrals using polylogarithms, providing closed-form expressions up to weight 4 and extending to Apéry-like sums with new identities and functional equations.

Contribution

The authors develop a novel identity linking generating functions of polylogarithms to sine integrals, enabling closed-form evaluations and generalizations to Apéry-like sums.

Findings

Closed-form expressions for log-sine integrals up to weight 4.

Extension of identities to Apéry-like sums and functional equations.

Derivation of hyperbolic analogues and new parametric identities.

Abstract

We present a new systematic method for evaluating generalized log-sine integrals in terms of polylogarithms. Our approach is based on an identity connecting ordinary generating functions of polylogarithms to integrals involving the sine function. This method provides closed-form expressions for log-sine integrals of weight up to 4 using only classical polylogarithms, while higher weights require Nielsen polylogarithms. Later we generalize this identity and show how it gives rise to numerous Ap\'ery-like formulae extending results of Koecher, Leshchiner and others. We also derive hyperbolic analogues and recover several functional equations between Nielsen polylogarithms. In the process, we derive new parametric identities similar to those given by Saha and Sinha.