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arXiv 2403.045552024-03math.COmath.CAmath.NT

$q$-类比的Wilf-Zeilberger种子与Ramanujan $1/π^k$-公式

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

Rheinische Friedrich-Wilhelms-Universität Bonn（波恩莱茵弗里德里希·威廉大学）

Kam Cheong Au

发表信息：: Adv. Math. 481 (2026), 111181
DOI：: 10.1016/j.aim.2026.111181

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AI总结：

本文提出Wilf-Zeilberger种子的概念，用于生成$q$-类比的Ramanujan型$1/π^k$公式，并揭示了部分非模形式的模伪模实例。

摘要：

我们发展了Wilf-Zeilberger种子的概念，作为生成WZ对和将经典超几何恒等式提升到$q$-设定的有力框架。作为应用，我们系统地获得了一个大规模的$ q $-类比的Ramanujan型$1/π^k$公式家族，扩展了此前仅限于孤立例子的文献。虽然大多数这些$ q $-类比是模形式的，但我们还揭示了几个奇特的模伪模实例。

英文摘要：

We develop the notion of Wilf-Zeilberger seeds as a powerful 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 body of literature previously limited to sporadic examples. While the majority of these $q$-analogues are modular, we also uncover several curious instances of mock modularity.

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