Pan-Xu conjecture and reduction formulas for polylogarithms

TL;DR

This paper proves the Pan-Xu conjecture related to generalized Mneimneh-like sums, introduces transformation and reduction formulas for multiple polylogarithms, and offers new representations for multiple zeta-star values.

Contribution

It generalizes and resolves the Pan-Xu conjecture, providing new reduction formulas and transformations for multiple polylogarithms and harmonic-star sums.

Findings

Resolved the Pan-Xu conjecture.

Derived new reduction formulas for multiple polylogarithms.

Presented new representations of multiple zeta-star values.

Abstract

The objective of the paper is the study of Mneimneh-like sums with a parametric variant of the multiple harmonic-star values. We generalize and resolve the Pan-Xu conjecture on generalized Mneimneh-like sums and present their transformation. As an application, we deduce new reduction formulas for specific multiple polylogarithms enabling lowering their depth, and provide additional findings on arithmetic means of multiple harmonic-star values, resulting in new representations of arbitrary multiple zeta-star values.