Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds

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.