A solution of Sun's $520 challenge concerning 520/pi

TL;DR

This paper proves a Ramanujan-type series for 520 divided by pi, using hypergeometric functions and modular parameters, building on recent work on Legendre polynomials and classical techniques for series for 1/pi.

Contribution

It provides the first proof of Sun's conjectured series for 520/pi, combining hypergeometric representations with modular methods.

Findings

Established a new series for 520/pi based on hypergeometric functions.

Connected recent Legendre polynomial generating functions with classical Ramanujan techniques.

Validated Sun's conjecture through rigorous mathematical proof.

Abstract

We prove a Ramanujan-type formula for $520/\pi$ conjectured by Sun. Our proof begins with a hypergeometric representation of the relevant double series, which relies on a recent generating function for Legendre polynomials by Wan and Zudilin. After showing that appropriate modular parameters can be introduced, we then apply standard techniques, going back to Ramanujan, for establishing series for $1/\pi$.