Talk: A solution of Sun's $520 challenge concerning 520/pi (SIAM)

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

Date: 2013/07/10
Occasion: SIAM Annual Meeting, Minisymposium on Symbolic Computation and Special Functions
Place: San Diego, CA


Series for \(1/\pi\) and their relation to modular forms have a long history going back to Ramanujan. We will review this connection and give a brief account of the history. We then indicate how to prove a Ramanujan-type formula for \(520/\pi\) that was conjectured by Zhi-Wei Sun. A key ingredient is a hypergeometric representation of the relevant double series, which relies on a recent generating function for Legendre polynomials by James Wan and Wadim Zudilin. For the purpose of the present minisymposium, a particular focus will be on the symbolic computation of singular moduli for modular functions such as, but not limited to, the classical \(j\)-function. Methods made possible by computer algebra as well as future challenges will be discussed. This talk is based on joint work with Mathew D. Rogers.


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