Computer-assisted construction of Ramanujan-Sato series for 1 over π.

Abstract

Referring to ideas of Sato and Yang in (Math Z 246:1-19, 2004) described a construction of series for 1 over $\pi$ starting with a pair (g, h), where g is a modular form of weight 2 and h is a modular function; i.e., a modular form of weight zero. In this article we present an algorithmic version, called "Sato construction". Series for $1/\pi$ obtained this way will be called "Ramanujan-Sato" series. Famous series fit into this definition, for instance, Ramanujan's series used by Gosper and the series used by the Chudnovsky brothers for computing millions of digits of $\pi$ . We show that these series are induced by members of infinite families of Sato triples $(N,{\gamma }_N,{\tau }_N)$ where $N>1$ is an integer and ${\gamma }_N$ a $2\times 2$ matrix satisfying ${\gamma }_N{\tau }_N=N{\tau }_N$ for ${\tau }_N$ being an element from the upper half of the complex plane. In addition to procedures for guessing and proving from the holonomic toolbox together with the algorithm "ModFormDE", as described in Paule and Radu in Int J Number Theory (17:713-759, 2021), a central role is played by the algorithm "MultiSamba", an extension of Samba ("subalgebra module basis algorithm") originating from Radu in (J Symb Comput 68:225-253, 2015) and Hemmecke in (J Symb Comput 84:14-24, 2018). With the help of MultiSamba one can find and prove evaluations of modular functions, at imaginary quadratic points, in terms of nested algebraic expressions. As a consequence, all the series for $1/\pi$ constructed with the help of MultiSamba are proven completely in a rigorous non-numerical manner.