A SOLUTION OF SUN'S $520 CHALLENGE CONCERNING $\frac{520}{\pi}$

Abstract

We prove a Ramanujan-type formula for 520/π conjectured by Zhi-Wei 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/π. Finally, we demonstrate that our approach can be used to also establish all further conjectures by Sun on series for 1/π of the same form.