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Wilf-Zeilberger seeds and non-trivial hypergeometric identities

Kam Cheong Au

TL;DR

The paper develops a seed-based Wilf-Zeilberger framework to prove challenging hypergeometric identities, including Ramanujan-type $1/\pi^k$ formulas and a $\zeta(5)$-series. It introduces WZ-seeds and a practical search strategy, guided by Gosper’s algorithm, to generate $F(n,k)$ with explicit WZ-mates and to control boundary terms. By coupling this machinery with CMZV theory and transformations of very-well-poised hypergeometric series, the authors extract CMZV coefficients and prove several new identities, such as $\sum_{n\ge0} 2^{-12n}\frac{(1/2)_n^7(1/4)_n(3/4)_n}{(1)_n^9}(43680n^4+\cdots+21)=\frac{2048}{\pi^4}$ and related formulas, while also addressing non-vanishing boundary term cases. The work provides a systematic route to verifying conjectures by Sun, Guillera, Zhao, and others, clarifies the CMZV/arithmetic content of these sums, and suggests both the reach and the current limits of the seed-WZ approach for hypergeometric identities.

Abstract

Through a systematic approach on generating Wilf-Zeilberger-pairs, we prove some hypergeometric identities conjectures due to Z.W. Sun, J. Guillera and Y. Zhao etc., including two Ramanujan-$1/π^4$, one $1/π^3$ formulas as well as a remarkable series for $ζ(5)$.

Wilf-Zeilberger seeds and non-trivial hypergeometric identities

TL;DR

The paper develops a seed-based Wilf-Zeilberger framework to prove challenging hypergeometric identities, including Ramanujan-type formulas and a -series. It introduces WZ-seeds and a practical search strategy, guided by Gosper’s algorithm, to generate with explicit WZ-mates and to control boundary terms. By coupling this machinery with CMZV theory and transformations of very-well-poised hypergeometric series, the authors extract CMZV coefficients and prove several new identities, such as and related formulas, while also addressing non-vanishing boundary term cases. The work provides a systematic route to verifying conjectures by Sun, Guillera, Zhao, and others, clarifies the CMZV/arithmetic content of these sums, and suggests both the reach and the current limits of the seed-WZ approach for hypergeometric identities.

Abstract

Through a systematic approach on generating Wilf-Zeilberger-pairs, we prove some hypergeometric identities conjectures due to Z.W. Sun, J. Guillera and Y. Zhao etc., including two Ramanujan-, one formulas as well as a remarkable series for .
Paper Structure (12 sections, 7 theorems, 171 equations, 3 figures)

This paper contains 12 sections, 7 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The central proposition and two motivating examples
  3. Wilf-Zeilberger seeds
  4. A source of WZ-pairs
  5. Computational implementation
  6. Computational implementation
  7. List of WZ-seeds
  8. Searching a suitable WZ-pair
  9. Vanishing of limit $\lim_{n\to\infty} \sum_{k\geq 0} F(n,k)$
  10. Euler sums and CMZVs
  11. Basic examples
  12. Examples with non-vanishing boundary term

Key Result

Proposition 2.1

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

Figures (3)

  • Figure 1: Plot of $\mathcal{E}_F(x)$, with $F$ in Example \ref{['ex_pi-4_1']}
  • Figure 2: Plot of $\mathcal{E}_F(x)$ for Example \ref{['sun_100_prob_1']}, maximum is attained at $x=(1+\sqrt{2})/2$
  • Figure 3: Plot of $\mathcal{E}_F(x)$ for Example \ref{['sun_100_prob_2']}, maximum is attained at $x=10/9$

Theorems & Definitions (35)

  • Proposition 2.1
  • proof
  • Example I
  • Example II
  • Definition 3.1
  • Theorem 3.3: Criterion of being WZ-seed
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 25 more