Reflection identities of harmonic sums of weight four

TL;DR

This paper derives and presents minimal reflection identities for harmonic sums of weight four, showing how products of simpler sums can generate more complex sums and providing a basis for constructing higher-order identities.

Contribution

It introduces a minimal set of bilinear reflection identities at weight four and explains how to derive higher-order identities using shuffle relations.

Findings

List of irreducible bilinear reflection identities at weight four

Method to construct trilinear and quartic identities from the bilinear set

Demonstration of pole structure decomposition in harmonic sums

Abstract

We consider the reflection identities for harmonic sums at weight four. We decompose a product of two harmonic sums with mixed pole structure into a linear combination of terms each having a pole at either negative or positive values of the argument. The pole decomposition demonstrates how the product of two simpler harmonic sums can build more complicated harmonic sums at higher weight. We list a minimal irreducible bilinear set of reflection identities at weight four which present the main result of the paper. We also discuss how other trilinear and quartic reflection identities can be easily constructed from our result with the use of well known shuffle relations for harmonic sums.