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

Referring to ideas of Sato and Yang in (Math Z 246:1-19, 2004) described a construction of series for 1 over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> starting with a pair (<i>g</i>, <i>h</i>), where <i>g</i> is a modular form of weight 2 and <i>h</i> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>π</mml:mi></mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>π</mml:mi></mml:math> . We show that these series are induced by members of infinite families of Sato triples <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:msub><mml:mi>γ</mml:mi> <mml:mi>N</mml:mi></mml:msub> <mml:mo>,</mml:mo> <mml:msub><mml:mi>τ</mml:mi> <mml:mi>N</mml:mi></mml:msub> <mml:mo>)</mml:mo></mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> is an integer and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>γ</mml:mi> <mml:mi>N</mml:mi></mml:msub> </mml:math> a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn> <mml:mo>×</mml:mo> <mml:mn>2</mml:mn></mml:mrow> </mml:math> matrix satisfying <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mi>γ</mml:mi> <mml:mi>N</mml:mi></mml:msub> <mml:msub><mml:mi>τ</mml:mi> <mml:mi>N</mml:mi></mml:msub> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:msub><mml:mi>τ</mml:mi> <mml:mi>N</mml:mi></mml:msub> </mml:mrow> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>τ</mml:mi> <mml:mi>N</mml:mi></mml:msub> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mi>π</mml:mi></mml:mrow> </mml:math> constructed with the help of MultiSamba are proven completely in a rigorous non-numerical manner.