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Chudnovsky-Ramanujan type formulae for non-compact arithmetic triangle groups
Journal
Journal of Number Theory
ISSN
0022314X
Date Issued
2022-12-01
Author(s)
Chen, Imin
Glebov, Gleb
Goenka, Ritesh
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 and give examples in higher class number, which may be of independent interest. We also prove some partial completeness results for Clausen type single summation series.
Volume
241
Publication link
Subjects