The Modularity of Z.-W. Sun's Conjectural Formulas for $\frac{1}{\pi }$

Abstract

In this work, we establish modular parameterizations for two general formulas for $\frac{1}{\pi }$ that subsume conjectural Ramanujan type formulas due to Z.-W. Sun, which have remained open since 2011. As an application of this, in a conceptual way we interpret how Sun's conjectural formulas arise and can be verified, as well as recover other cases that were proved by Cooper, Wan and Zudilin.