Chudnovsky-Ramanujan Type Formulae for non-Compact arithmetic triangle groups

TL;DR

This paper introduces a systematic method to derive Ramanujan-type series for non-compact arithmetic triangle groups, expanding known series and discovering new ones, with an algorithm for confirming singular values of Eisenstein series.

Contribution

It develops a uniform approach to derive Ramanujan-type series for non-compact arithmetic triangle groups, including new series not previously identified.

Findings

Derived all rational Ramanujan-type series for levels 1-4 by Chan-Cooper

Discovered two additional rational series first found by Z.-W. Sun

Provided an algorithm to confirm singular values of Eisenstein series

Abstract

We develop a uniform method to derive Chudnovsky-Ramanujan type formulae for triangle groups based on a generalization of a method of Chudnovsky and Chudnovsky; in particular, we carry out the method systematically for non-compact arithmetic triangle groups and one non-Fuchsian covering. As a result, we derive all rational Ramanujan type series given by Chan-Cooper for levels 1-4, as well as two additional rational series of a similar form prescribed by Chan-Cooper for these levels, but not found in the paper of Chan-Cooper. These two additional series were first found by Z.-W. Sun in a slightly different form. We also derive additional rational series of a similar form, but not found in the papers of Chan-Cooper nor Z.-W. Sun. As an ingredient in the method, we give an algorithm to rigorously confirm the singular values of normalized Eisenstein series of weight 2, which may be of independent interest.