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Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds

Kam Cheong Au

TL;DR

This work extends the Wilf-Zeilberger framework by using WZ-seeds, a parametric generalization that yields families of WZ-pairs from which coefficients are extracted to produce hypergeometric series with harmonic-number factors. The authors obtain a rapidly convergent series for the depth-two MZV $ζ(5,3)$ and conjecture a simple Hilbert-Poincaré function for the coefficient spaces, suggesting an underlying graded-algebra structure with five generators. Across explicit constructions and numerous examples, they connect parameter-extracted series to MZVs, Ramanujan-type $1/π$ formulas, and broader algebraic structures, highlighting both rigorous identities and conjectural relations. The framework provides a systematic route to discover and organize new hypergeometric-harmonic identities with potential deepening insights into the algebraic organization of MZVs and related constants.

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 $ζ(5,3)$, 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.

Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds

TL;DR

This work extends the Wilf-Zeilberger framework by using WZ-seeds, a parametric generalization that yields families of WZ-pairs from which coefficients are extracted to produce hypergeometric series with harmonic-number factors. The authors obtain a rapidly convergent series for the depth-two MZV and conjecture a simple Hilbert-Poincaré function for the coefficient spaces, suggesting an underlying graded-algebra structure with five generators. Across explicit constructions and numerous examples, they connect parameter-extracted series to MZVs, Ramanujan-type formulas, and broader algebraic structures, highlighting both rigorous identities and conjectural relations. The framework provides a systematic route to discover and organize new hypergeometric-harmonic identities with potential deepening insights into the algebraic organization of MZVs and related constants.

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 , 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.
Paper Structure (4 sections, 1 theorem, 100 equations, 6 tables)

This paper contains 4 sections, 1 theorem, 100 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Wilf-Zeilberger seeds and pairs
  3. Explicit Examples
  4. More Examples

Key Result

Proposition 2.2

au2025wilf Suppose $F,G: \mathbb{N}^2\to \mathbb{C}$ are two functions such that such that then $\lim_{n\to \infty} \sum_{k\geq 0} F(n,k)$ exists and is finite, also

Theorems & Definitions (26)

  • Conjecture 1.1
  • Definition 2.1
  • Proposition 2.2
  • Example I
  • Conjecture 3.1
  • Conjecture 3.2
  • Remark 3.3
  • Example II
  • Conjecture 3.4
  • Example III
  • ...and 16 more