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

Mark van Hoeij, Wei-Lun Tsai, Dongxi Ye

Abstract

In this work, we establish modular parameterizations for two general formulas for $\frac{1}π$ 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.

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

Abstract

In this work, we establish modular parameterizations for two general formulas for 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.

Paper Structure

This paper contains 7 sections, 6 theorems, 68 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Modular parameterizations
  3. CM points and evaluations
  4. CM points $\tau_{\pm}$
  5. Evaluations of $B(\tau)$
  6. Evaluations of $\frac{Y(\tau_{+})}{Y(\tau_{-})}$
  7. Remaining conjectures

Key Result

Theorem 1.1

The equalities given in Conjecture sunconj all hold.

Theorems & Definitions (17)

  • Conjecture 1.1: Z.-W. Sun S11
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2: Cooper--Wan--Zudilin CWZ
  • Lemma 2.1
  • proof
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 7 more