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$q$-analogues of Wilf-Zeilberger seeds and Ramanujan $1/π^k$-formulas

Kam Cheong Au

TL;DR

The paper develops Wilf-Zeilberger seeds as a systematic framework to generate $q$-Wilf-Zeilberger pairs and translate classical hypergeometric identities into $q$-analogues of Ramanujan-type $1/\pi^k$-formulas. By applying this to a broad set of seeds, it produces 21 new $q$-analogues, most modular but three exhibiting mock modularity, and it clarifies how accessory parameters and limiting processes yield multi-parameter families. It also discusses analytic subtleties and limitations, including formulas not amenable to WZ-style proofs, and highlights potential extensions to $p$-adic analogues and deeper modular objects. Overall, the work provides a practical dictionary of seeds and a scalable workflow for deriving and validating $q$-analogues with broad arithmetic structure and potential applications beyond the modular setting.

Abstract

We develop the notion of Wilf-Zeilberger seeds as a natural framework for generating WZ-pairs and for lifting classical hypergeometric identities to the $q$-setting. As an application, we systematically obtain a large family of $q$-analogues of Ramanujan-type $1/π^k$ formulas, extending a literature previously limited to sporadic examples. While the majority of these $q$-analogues are modular, we also uncover several curious instances of mock modularity.

$q$-analogues of Wilf-Zeilberger seeds and Ramanujan $1/π^k$-formulas

TL;DR

The paper develops Wilf-Zeilberger seeds as a systematic framework to generate -Wilf-Zeilberger pairs and translate classical hypergeometric identities into -analogues of Ramanujan-type -formulas. By applying this to a broad set of seeds, it produces 21 new -analogues, most modular but three exhibiting mock modularity, and it clarifies how accessory parameters and limiting processes yield multi-parameter families. It also discusses analytic subtleties and limitations, including formulas not amenable to WZ-style proofs, and highlights potential extensions to -adic analogues and deeper modular objects. Overall, the work provides a practical dictionary of seeds and a scalable workflow for deriving and validating -analogues with broad arithmetic structure and potential applications beyond the modular setting.

Abstract

We develop the notion of Wilf-Zeilberger seeds as a natural framework for generating WZ-pairs and for lifting classical hypergeometric identities to the -setting. As an application, we systematically obtain a large family of -analogues of Ramanujan-type formulas, extending a literature previously limited to sporadic examples. While the majority of these -analogues are modular, we also uncover several curious instances of mock modularity.
Paper Structure (6 sections, 4 theorems, 183 equations, 1 figure)

This paper contains 6 sections, 4 theorems, 183 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Wilf-Zeilberger seeds
  3. Definition and basic properties
  4. List of WZ-seeds
  5. Two introductory examples
  6. $q$-analogue of Ramanujan-type $1/\pi^k$ formulas: 21 examples

Key Result

Proposition 1.2

The (classical or $q$-) hypergeometric term $f(a_1,\cdots,a_m,k)$ is a WZ-seed if and only if for all $A_1,\cdots,A_m, K\in \mathbb{Z}$, has a hypergeometric WZ-mate $G(n,k)$, that is, $F(n+1,k) - F(n,k) = G(n,k+1) - G(n,k).$

Figures (1)

  • Figure 1: A schematic illustration of the three concepts: WZ-seeds, WZ-pair and hyoergeometric summation identity, for both classical and $q$-version.

Theorems & Definitions (35)

  • Definition 1.1
  • Proposition 1.2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof : Proof sketch
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 25 more