$q$-analogues of Wilf-Zeilberger seeds and Ramanujan $1/{\pi }^k$-formulas

Abstract

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 $1/{\pi }^k$ 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.