Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds

TL;DR

This paper introduces a novel method using Wilf-Zeilberger pairs to discover hypergeometric series involving harmonic numbers, including a new series for a specific multiple zeta value, and explores algebraic structures related to WZ-seeds.

Contribution

It presents a new technique for deriving hypergeometric series with harmonic numbers and provides the first known series for $ ext{zeta}(5,3)$, along with conjectures about algebraic structures behind WZ-seeds.

Findings

Discovered a rapidly convergent series for $ ext{zeta}(5,3)$.

Experimented with Hilbert-Poincare series linked to WZ-seeds.

Conjectured a simple form for the Hilbert-Poincare series indicating underlying algebraic structure.

Abstract

By extracting coefficients from Wilf-Zeilberger pairs with respect to auxiliary parameters, we discover many nontrivial hypergeometric series involving harmonic numbers. In particular, we obtain a rapidly convergent series for the depth-two multiple zeta value $\zeta(5,3)$, which appears to be the first result of its kind in the literature. We also experiment with the Hilbert-Poincare series attached with a WZ-seed and conjecture that it admits a remarkably simple form, suggesting the presence of an underlying graded algebra structure behind WZ-seeds.