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

TL;DR

This paper establishes modular parameterizations for Z.-W. Sun's conjectural formulas for 1/π, providing a conceptual framework that explains their origin and verifies their validity, extending previous results by other mathematicians.

Contribution

It introduces modular parameterizations for Sun's conjectural formulas for 1/π, unifying and explaining their structure and relation to known results.

Findings

Established modular parameterizations for Sun's formulas

Provided a conceptual interpretation of Sun's conjectures

Recovered previously proved cases by other researchers

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.