[bib]
@techreport{RISC7134,
author = {Ralf Hemmecke and Peter Paule and Cristian-Silviu Radu},
title = {{Computer-assisted construction of Ramanujan-Sato series for 1 over pi}},
language = {english},
abstract = {Referring to ideasof Takeshi Sato, Yifan Yang in~cite{YangDE} described a construction ofseries for $1$ over $pi$ startingwith a pair $(g,h)$, where $g$ is a modular formof weight $2$ and $h$ is a modular function; i.e.,a modular form of weight zero. In this article we present an algorithmicversion,called ``Sato construction''. Series for $1/pi$ obtained this way will becalled ``Ramanujan-Sato''series. Famous series fit into this definition, for instance, Ramanujan'sseries used by Gosperand the series used by the Chudnovsky brothersfor computing millions of digits of $pi$. Weshow that these series are induced by membersof infinite families of Sato triples $(N, gamma_N,tau_N)$ where $N>1$ is an integer and $gamma_N$ a $2times 2$ matrixsatisfying $gamma_N tau_N=N tau_N$ for$tau_N$ being an element from the upper half of thecomplex plane.In addition to procedures for guessingand proving from the holonomic toolbox togetherwiththe algorithm ``ModFormDE'', as describedin~cite{PPSR:ModFormDE1}, a central roleis played by the algorithm ``MultiSamba'',an extension ofSamba (``subalgebra module basis algorithm'') originating fromcite{Radu_RamanujanKolberg_2015} and cite{Hemmecke}.With thehelp of MultiSamba one canfind 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 withthe help of MultiSamba are proven completelyin a rigorous non-numerical manner.},
number = {25-01},
year = {2025},
month = {January},
keywords = {modular forms and functions, holonomic differential equations, Ramanujan-Sato series for 1 over pi, MultiSamba algorithm},
length = {58},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
