Computer-assisted construction of Ramanujan-Sato series for 1 over pi

Referring to ideas of Takeshi Sato, Yifan Yang in~cite{YangDE} 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 $2times 2$ 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~cite{PPSR:ModFormDE1}, a central role is played by the algorithm ``MultiSamba'', an extension of Samba (``subalgebra module basis algorithm'') originating from cite{Radu_RamanujanKolberg_2015} and cite{Hemmecke}. 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.